Piecewise Functions




This section covers:

Introduction to Piecewise Functions

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Piecewise functions are just what they are named: pieces of different functions all on one graph.  The easiest way to think of them is if you drew more than one function on a graph, and you just erased parts of the functions where they aren’t supposed to be (along the x’s); they are defined differently for different intervals of x.  So y is defined differently for different values of x; we use the x to look up what interval it’s in, so we can find out what the y is supposed to be.

Here’s an example and graph:

Piecewise Function Example and Graph

So what this means is for every x less than or equal to –2, we need to graph the line 2x + 8, as if it were the only function on the graph.  For every x value greater than –2, we need to graph Bold X Squared, as if it were the only function on the graph.  Then we have to “get rid of” the parts that we don’t need.  Remember that we still use the origin as the reference point for both graphs!

See how the vertical line x = 2 acts as a “boundary” line between the two graphs?

Note that the point (–2, 4) has a closed circle on it.  Technically, it should only belong to the 2x + 8 function, since that function has the less than or equal sign, but since the point is also on the Bold X Squared graph, we can just use a closed circle as if it appears on both functions.

See, not so bad, right?

Evaluating Piecewise Functions

Sometimes, you’ll be given piecewise functions and asked to evaluate them; in other words, find the y values when you are given an x value.  Let’s do this for x = –6 and x = 4 (without using the graph).   Here is the function again:

Piecewise Function Example

We first want to look at the conditions at the right first, to see where our x is.  When x = –6, we know that it’s less than –2, so we plug in our x to 2x + 8 only.  So f(x) or y is (2)(–6) + 8, or –4.   We don’t even care about the Bold X Squared!  It’s that easy.  You can also see that we did this correctly by using the graph above.

Now try x = 4.  We look at the right first, and see that our x is greater than –2, so we plug it in the x Squared.  (We can just ignore the 2x + 8 this time.)  So f(x) or y is 42 = 16.

Graphing Piecewise Functions

You’ll probably be asked to graph piecewise functions.  Sometimes the graphs will contain functions that are non-continuous or discontinuous, meaning that you have to pick up your pencil in the middle of the graph when you are drawing it (like a jump!).   Continuous functions means that you never have to pick up your pencil if you were to draw them from left to right.

And remember that the graphs are true functions only if they pass the Vertical Line Test.

Let’s draw these piecewise functions and determine if they are continuous or non-continuous.  Note how we draw each function as if it were the only one, and then “erase” the parts that aren’t needed.  We’ll also get the Domain and Range like we did here in the Algebraic Functions section.

Piecewise Continuous and NonContinuous

We can actually put piecewise functions in the graphing calculator:

Piecewise Functions in the Graphing Calculator

How to Tell if Piecewise Function is Continuous or Non-Continuous

To tell if a piecewise graph is continuous or non-continuous, you can look at the boundary points and see if the y point is the same at each of them.  (If the y’s were different, there’d be a “jump” in the graph!)

Let’s try this for the functions we used above:

Checking to see if Piecewise Functions are Continuous

Obtaining Equations from Piecewise Function Graphs

You may be asked to write a piecewise function, given a graph.  Now that we know what piecewise functions are all about, it’s not that bad!

To review how to obtain equations from linear graphs, see Obtaining the Equations of a Line, and from quadratics, see Finding a Quadratic Equation from Points or a Graph.

Here are the graphs, with explanations on how to derive their piecewise equations:

Deriving Piecewise Function Equations from Graphs

Absolute Value as a Piecewise Function

We can write absolute value functions as piecewise functions – it’s really cool!   You might want to review Algebraic Equations with Absolute Value before continuing on to this topic.

Let’s say we have the function Absolute Value of x.  From what we learned earlier, we know that when x is positive, since we’re taking the absolute value, it will still just be x.  But when x is negative, when we take the absolute value, we have to take the opposite (negate it), since the absolute value has to be positive.  Make sense?  So, for example, if we had |5|, we just take what’s inside the absolute sign, since it’s positive.  But for |–5|, we have to take the opposite (negative) of what’s inside the absolute value to make it 5 (– – 5 is 5).

This means we can write this absolute value function as a piecewise function.  Notice that we can get the “turning point” or “boundary point” by setting whatever is inside the absolute value to 0.  

For example, we can write  Absolute Value as Piecewise Function.

Also note that, if the function is continuous (there is no “jump”) at the boundary point, it doesn’t matter where we put the “less than or equal to” (or “greater than or equal to”) signs, as long as we don’t repeat them!   We can’t repeat them because, theoretically, we can’t have 2 values of y for the same x, or we wouldn’t have a function.

Here are more examples, with explanations.  (You might want to review Quadratic Inequalities for the second example below):

Deriving Piecewise Functions from Absolute Value Functions

You may also be asked to take an absolute value graph and write it as a piecewise function:

Obtaining Piecewise Functions from Graphs

Transformations of Piecewise Functions

Let’s do a transformation of a piecewise function.

We learned how about Parent Functions and their Transformations here in the Parent Graphs and Transformations section.   You’ll probably want to read this section first, before trying a piecewise transformation.

Let’s transform the following piecewise function flipped around the x axis, vertically stretched by a factor of 2 units, 1 unit to the right, and 3 units up.

So we will draw , \(-2f\left( x-1 \right)+3\) where:

Piecewise Function to Transform

Let’s make sure we use the “boundary” points when we fill in the t-chart for the transformation.   Remember that the transformations inside the parentheses are done to the x (doing the opposite math), and outside are done to the y.   So to come up with a t-chart, as shown in the table below, we can use key points, including two points on each of the “boundary lines”.

Note that because this transformation is complicated, we can come up with a new piecewise function by transforming the 3 “pieces” and also transforming the “x”s where the boundary points are (adding 1, or going to the right 1), since we do the opposite math for the “x”s:

Transformed Piecewise Function

Here are the “before” and “after” graphs, including the t-chart:

Transformed Piecewise Function Graphs

Piecewise Function Word Problems

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Problem:

Your favorite dog groomer charges according to your dog’s weight.  If your dog is 15 pounds and under, the groomer charges $35.  If your dog is between 15 and 40 pounds, she charges $40.  If your dog is over 40 pounds, she charges $40, plus an additional $2 for each pound.

(a)   Write a piecewise function that describes what your dog groomer charges.
(b)   Graph the function.
(c)   What would the groomer charge if your cute dog weighs 60 pounds?

Solution:

(a)   We see that the “boundary points” are 15 and 40, since these are the weights where prices change.  Since we have two boundary points, we’ll have three equations in our piecewise function.  We have to start at 0, since dogs have to weigh over 0 pounds:  Word Problem Boundary

We are looking for the “answers” (how much the grooming costs) to the “questions” (how much the dog weighs) for the three ranges of prices.  The first two are just flat fees ($35 and $40, respectively).  The last equation is a little trickier; the groomer charges $40 plus $2 for each pound over 40.  Let’s try real numbers: if your dog weighs 60 pounds, she will charge $40 plus $2 times 20 (60 – 40).  We’ll turn this into an equation: 40 + 2(x – 40), which simplifies to 2x – 40 (see how 2 is the slope?).   So the whole piecewise function is:

Word Problem Piecewise Function

(b)   Let’s graph:

Graph of Dog Word Problem

Note that this piecewise equation is non-continuous.

Also note a reasonable domain for this problem might be Domain (given dogs don’t weigh over 200 pounds!) and a reasonable range might be Range.

(c)   If your dog weighs 60 pounds, we can either use the graph, or the function to see that you would have to pay $80.  Whoa!  That costs more than a human haircut (at least my haircuts)!

Problem:

You plan to sell She Love Math t-shirts as a fundraiser.  The wholesale t-shirt company charges you $10 a shirt for the first 75 shirts.  After the first 75 shirts you purchase up to 150 shirts, the company will lower its price to $7.50 per shirt.  After you purchase 150 shirts, the price will decrease to $5 per shirt.  Write a function that models this situation.

Solution:

We see that the “boundary points” are 75 and 150, since these are the number of t-shirts bought where prices change.  Since we have two boundary points, we’ll have three equations in our piecewise function.  We’ll start with x greater than or equal to 1, since, we assume at least one shirt is bought.  Note in this problem, the number of t-shirts bought (x), or the domain, must be a integer, but this restriction shouldn’t affect the outcome of the problem.

T-Shirt Word Problem

We are looking for the “answers” (total cost of t-shirts) to the “questions” (how many are bought) for the three ranges of prices.

For up to and including 75 shirts, the price is $10, so the total price would 10x.  For more than 75 shirts but up to 100 shirts, the cost is $7.50, but the first 75 t-shirts will still cost $10 per shirt.  So the second function includes the $750 spent on the first 75 shirts (75 times $10), and also includes $7.50 times the number of shirts over 75, which would be (x – 75).   For example, if you bought 80 shirts, you’d have to spend $10 times 75 = $750, plus $7.50 times 5 (80 – 75) for the shirts after the 75th shirt.

Similarly, for over 150 shirts, we would still pay the $10 price up through 75 shirts, the $7.50 price for 76 to 150 shirts (75 more shirts), and then $5 per shirt for the number of shirts bought over 150.  So we’ll pay 10(75) + 7.50(75) + 5(x – 150) for x shirts.  Put in numbers and try it!

So the whole piecewise function is:

Word Problem Piecewise Function

Problem:

What value of a would make this piecewise function continuous?

Make a Piecewise Function Continuous

Solution:

For the piecewise function to be continuous, at the boundary point (where the function changes), the two y values must be the same.   So we can plug in –2 for x in both of the functions and make sure the y’s are the same

Solving for a

If a = 26, the piecewise function is continuous!

Learn these rules, and practice, practice, practice!


Click on Submit (the arrow to the right of the problem) to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.

If you click on “Tap to view steps”, you will go to the Mathway site, where you can register for the full version (steps included) of the software.  You can even get math worksheets.

You can also go to the Mathway site here, where you can register, or just use the software for free without the detailed solutions.  There is even a Mathway App for your mobile device.  Enjoy!

On to Matrices and Solving Systems with Matrices – you are ready!

117 thoughts on “Piecewise Functions

    • Kiko,
      Thank you so so much for taking the time to write, and I’m so glad the site is helpful. This makes me want to keep writing more and more!
      Thanks again,
      Lisa

    • Thank you for these postings. It is an amazing website. I plan to use some of your examples in my class. Hope you will do this for trigonometry and calculus too.

  1. Great post and website, but I have a warning about your graphing calculator example:
    The graphing calculator evaluates inequalities as either 0 (for false) or 1 (as true). So, in the example above, when you write your y1 = (x+4)(x<1), for all x-values less than 1, the graph appears because (x+4)*1 is the diagonal line, but for x-values more than 1, what is generated is (x+4)*0=0, creating a horizontal line on the x-axis. You luck out in your example above since the y=0 line overlaps the axis, so it is "hidden", but using the trace command, the cursor will show it.
    I recommend instead of multiplying by the inequality, divide by it. Then, y1 = (x+4)/(x<1), and a true inequality will simply be division by 1, meaning no change, but a false inequality will be division by zero, creating a failure and thus nothing is graphed.
    Thank you for the great explanation and overview, and I do intend to use this in class!

    • Brad,
      Thank you so much for pointing this out; I am going to change it per your suggestion! This clears up some confusion I had with that example!
      Thanks again. Please let me know if you find anything else 😉
      Lisa

  2. How do I approach/solve this math problem? Pls solve or mail a solution. Thanks

    MATHEMATICAL REASONING EXERCISE
    Instructions: The following problem requires you to decide which job you will accept among three choices. There is no “correct” answer. You will decide which job is the best one for you from a financial standpoint and you will be graded on the correctness of your equations and on the clarity, accuracy and soundness of your reasoning. We will assume that all three jobs mentioned are equally appealing to you in all other aspects except those described in the problem. Also, for simplicity, it is assumed that no taxes or other deductions will apply to your monthly check. Your answer must be written in complete and grammatically correct sentences. You may use graphs and/or the equations you derive to justify your arguments.

    PROBLEM – CHOOSING THE BEST JOB
    You are a college freshman and have planned your academic career carefully. You will complete college and receive your Bachelor’s degree in five years. You were fortunate to have found a part-time job that is guaranteed for the five years and that will allow you to pay your bills while in school. We will call this Job A. After working at Job A for 25 months you are offered two other jobs at the same time, Job B and Job C, both of which are guaranteed for the remaining time (35 months) you are in school. You must make a decision whether to stay with Job A or switch to Job B or Job C. The decision must be made right after the 25th month; it cannot be delayed.
    Job A: This job pays $1850 per month plus a bonus of $250 per month. However, you will not receive the bonus on a monthly basis. It will be placed into an account (paying no interest). After you have worked for the company for 30 months you will receive, in a lump sum, all the money in the account at that time. If you quit before 30 months, you will receive none of the bonus money. Then, beginning with the 31st month, the $250 bonus will again be placed into the savings account each month for the remaining time you work for the company. At the end of 60 months, you will receive, in a lump sum, the remaining bonus money. If you quit before 60 months, you will receive none of the bonus money that accumulated beginning with the 31st month.
    Job B: This job pays $2250 per month, but offers no bonus.
    Job C: This job pays $1700 per month and offers a one-time bonus of $21,250 at the end of the 35 months.
    Remember that your decision is to be made when you have worked at Job A for exactly 25 months.

    YOUR ASSIGNMENT
    1. For each job, give the equation (function) that calculates the amount A of money earned after m months. Your answer should involve piecewise-functions.
    2. Explain the advantages of each job from a financial standpoint. Remember that you have only worked at Job A for 25 months, and will receive no bonus from that job if you quit now.
    3. State which job you believe is the best one for you and explain, in a logical and convincing way, the reasons why. There is no correct answer, but you must support your decision in a logical manner, and acknowledge the results of your mathematical analysis.

    • Great problem!! Let me give you a start and see if you can get going from there.

      If you were planning to switch to job B after 25 months (the Job A – Job B route), I believe the function would be: f(m) = 1850m, for 0 <= m <= 25 (because you would be doing job A for 25 months first and you make 1850 a month). Then you would make 2250 for any month AFTER the first 25 months, up to 60 months. So this would be 1850(25) + 2250(m - 25), for 25 < m <= 60. (Try this by plugging in 26 for m, and you'll see how it works). You wouldn't have to worry about any bonus. (You should simplify this equation). If you stay at Job A, I believe it would be: f(m) = 1850m, for 0 <= m < 30 (since if you quit, you don't get the any bonus), (1850 + 250)(30) + 1850 (m - 30), for 30 <= m < 60 (since you'd get a bonus for the first 30 months, but not another one unless you work a full 60 months), and (1850 + 250)m, for m = 60 (since you'd get a bonus all the way through). If you went the Job A-Job C route, the function would be f(m) = 1850m, for 0 <= m <= 25 (since you're at Job A for the first 25 months), 25(1850) + 1700 (m - 25) for 25 < m = 60 (since you make 1700 a month for the months after month 25), and 25(1850) + (1700)(35) +21250 for m = 60 (since you'd only get a bonus if you work until month 60). Does this make sense?

        • I went ahead and updated the answer a little; here it is again. Does it make sense? Lisa
          If you were planning to switch to job B after 25 months (the Job A – Job B route), I believe the function would be: f(m) = 1850m, for 0 <= m <= 25 (because you would be doing job A for 25 months first and you make 1850 a month). Then you would make 2250 for any month AFTER the first 25 months, up to 60 months. So this would be 1850(25) + 2250(m – 25), for 25 < m <= 60. (Try this by plugging in 26 for m, and you’ll see how it works). You wouldn’t have to worry about any bonus. (You should simplify this equation). If you stay at Job A, I believe it would be: f(m) = 1850m, for 0 <= m < 30 (since if you quit, you don’t get the any bonus), (1850 + 250)(30) + 1850 (m – 30), for 30 <= m < 60 (since you’d get a bonus for the first 30 months, but not another one unless you work a full 60 months), and (1850 + 250)m, for m = 60 (since you’d get a bonus all the way through). If you went the Job A-Job C route, the function would be f(m) = 1850m, for 0 <= m <= 25 (since you’re at Job A for the first 25 months), 25(1850) + 1700 (m – 25) for 25 < m = 60 (since you make 1700 a month for the months after month 25), and 25(1850) + (1700)(35) +21250 for m = 60 (since you’d only get a bonus if you work until month 60). Does this make sense?

          • Don’t we need to have a number for m to plug in for f(m)? How would we determine the number?

          • Oh sorry – I didn’t read the actual assignment. Let’s use m = 60 (assuming we will work the whole 5 years) and see which gives us the most money: For only job A, it will be $126000, for job A-B, it would be $125000, and for the job A-C route, it would be $127000. So it looks like the A-C route would be the best, if I did the problem correctly. Lisa

  3. Can you help me with this problem? I have the values set up but I can’t figure out the equations needed in the function.
    You plan to sell t-shirts as a fundraiser. The company you wish to purchase your t-shirts from charge $10 per shirt for the first 75 shirts. After the first 75 shirts you purchase up to 150 shirts, the company will lower its price to $7.50 per shirt. After you purchase 150 shirts, the price will decrease to $5 per shirt. Write a function that models this situation.

    • I would model this with a piecewise function: f(x) = 10x for 0< = x<=75, 10(75) + 7.5(x - 75) for 75 150. Does this make sense? Lisa

  4. Excellent examples. I need a quick site for my students to see Domain and Range for piecewise functions. When I googled you were one of the first sites to pop up. I am so glad that I ran across your site and will be bookmarking it for further use. Thanks for your wonderful work.

    • Thank you so much for writing! This makes me want to keep going on this site – I have a lot of work to do. Thanks again 🙂 Lisa

  5. I just want to say thanks to the person who took his/her time to do all this wonderful posting. I had in the past taken calculus class but let it go because I was not getting it. Reading the posts on this website, makes me fell like going back to school.
    Again thanks for the wonderful job.

  6. how do I answer this?

    A sales company advertises a job opening. The annual salary is to be $10000 plus commission. The commission is paid at the rate of 8% on all sales up through $100,000. For sales that exceed $100,000, the commission rate is 12%

    • Here’s what I got for this one. The best way to do these is to put real numbers in, and then see what you are doing with the math. Here’s what a yearly compensation would be:
      y = 10000 + .08x, for 0 < = x <= 100000; 10000 + .08(100000) + (x - 100000).12, for x > 100000.

      Does that make sense? You’d write it with brackets like you see with piecewise functions. You should probably simplify it too for the second one – if you teacher wants that.
      Lisa

      • i would like to ask a question… this is our assignment,… what we are going to do is to determine if the problem is a function and we need to explain… here are the problem as follow:
        1. the average speed of a car in kilometers per hour.
        2.the area of the polygon is 1/2 the product of its base by its altitude.
        3. the interest earned if P=20,000, r=5% and t=3years.
        4. y is greater than or equal to 5, y is natural number.

        i hope you can help me… thank you so much

        • Hi and thanks for your question!
          Think of a function of only having ONE ANSWER to a question (ONE y for each x). So for 1), it is a function, since there’s only one average speed of a car (one y) for each car (x). 2) is a function since each polygon (x) only has ONE area (y). 3) is a function since you only have one amount of interest earned (y) for those parameters. 4) is not a function since there can be two y’s for one x (graph it to see). Does this make sense?
          Lisa

          • hi,
            again I would to ask a question. what we are going to do is to determine again if it is a function or not and explain. my problem is, I don’t know how to explain very well.

            1. y=x^2 + 3x + 1
            2. { (x,y) / y^4 =x
            3. { (x,y) / y= squareroot of 2x-3
            4. y= -3x + 2
            5. {(x,y) / y= (1-x)/(x+1)
            6. { y< x, x=3
            7.{ (x,y) / ^3= x+1
            8.{(x,y) / y^2 = x-3
            9. y is less than or equal to 0, y is a whole number
            10. {(x,y) / y= |2x|
            11. { y= x/1-x

          • Sure – no problem. 1) yes a function since it’s a vertical parabola and it passes the vertical line test (no 2 points on the same vertical line), 2) Not a function since it doesn’t pass vertical line test, 3) Yes a function – passes vertical line test, 4) yes a function, straight line (pass VLT), 5) Yes a function – passes vertical line test, 6) Not a function – many points can be found on the same vertical line, 7) Not sure if you mean y^3 – if so, yes, it’s a function, since it’s a cube root function, 8) Not a function – sideways parabola and doesn’t pass vertical line test, 9) Not a function – many points can be found on the same vertical line, 10) Yes a function – passes vertical line test, and 11) Yes a function – passes vertical line test.
            Hope that helps! Lisa

  7. Hello,

    I recently discovered this site. It’s great! (except that we should be selling She Loves Math T-shirts instead of She Love Math ones.) I was wondering if you could add a part on this page about transforming/translating piecewise functions. I understand how to translate functions, but I am confused about when to change the rules for the equation when transforming piecewise functions. Thanks!

    • Thanks so much for writing!! Yes, I’d be happy to add a piecewise function section to the Transformations section – I’m working on transforming trig functions next, but I’ll try to squeeze it in. Great suggestion!! Lisa (by the way, do you make/sell t-shirts? – I want to start to have them available on the site).

      • Thank you so much! I really appreciate it. (unfortunately, I don’t sell t-shirts, I just noticed the little typo). I am trying to teach myself Algebra 2 this summer so that I can skip the course and take precalculus next year. I have been using Thinkwell’s Alg. 2 course, which is a pretty good course, except that I sometimes have to search online if I have questions. What I noticed that most instructors forget to mention is that when you transform a piecewise functions the intervals or “boundaries” only change when transforming the equation horizontally (either shifting, stretching, or compressing it). In order to find what the new boundary is, you must plug the transformation into the x value and solve the inequality for x. At least this is the conclusion I came to after searching for the last two days. I couldn’t find this written on any website! Maybe this is supposed to be intuitive, but it wasn’t obvious to me at first. It would be great if this site explained that. Thanks!

  8. I appreciate your work on this website. Sometimes I find that when my students struggle with a certain topic, a different explanation of it can be helpful. I return to your site often to see you explain various topics and use your ideas to help reach the students that continue to struggle. Thanks for your work and helpful information.

    • Thank you so much! Your math blog is amazing – I love it. May I link from my site?
      Thanks again for writing; it makes me want to finish the site faster 😉
      Lisa

  9. I have a question. I believe I know the answer but I don’t know how to show it algebraically.

    Could you help me please?

    f(x)=sq rt(2x-1)-3/(x-5) It is defined as a piecewise function.

    F(x)={ f(x), x doesn’t =5
    { b , x=5

    Find the value of b that will make the function continuous?

    I would really appreciate a response. The answer is not as important as the process.

    Thank you
    Candace

  10. Hello. I have an assignment and I’m quite confused since it involves two absolute values.
    “Define f(x)= |x-1| – |x+1| without absolute value bars piecewise in the following interval:
    (-infinity,-1), [-1,1), [1,+infinity)” I’ll really appreciate if you could help me out with explanations. Great website by the way! 🙂

    • Thanks for writing! This is a tricky one, but, for each interval, look to see if |x – 1| and |x + 1| are positive or negative. If they are positive, just take off absolute value and use as regular expression. If they are negative, take off absolute value, but make everything in the absolute value negative, since the absolute value would be the opposite of the expression.
      So for (-infinity, -1), we would have (1-x)-(x-1) = 1-x-x+1 = 2. For [-1,1), (x-1) is negative, but (x+1) is positive, so we have (1-x) – (x+1) = 1-x-x-1 = -2x. For [1, infinity), both are positive, so we just take off absolute value signs: (x-1)-(x+1) = x-1-x-1 = -2.
      So f(x) = 2, for (-infinity, -1), -2x for [-1, 1), and -2 for [1, infinity). Does that make sense? Lisa

  11. Some of your problems are beautifully done. However, your elucidation on transformation of piecewise functions is terrible. No proper explanation is given on the T. Chart. A lot more explanation would be of the essence.

    • Harry,
      Thanks so much for writing. I reread that portion of the Piecewise Function section and agree that it’s confusing. I’m in the process of cleaning it up and also adding more content in the Parent Function and Transformations section. Let me know if you find any other examples of confusing information on the site. Thanks again! Lisa

  12. I have a question reagarding your function for the problem regarding the t-shirts. Why is the condition for 10x 0<x<=75 if we assume that we buy AT LEAST ONE shirt? If x is greater than 0, then we leave room for a possibility that the number of shirts can be 0.4 or any decimal number. Wouldn't it be more realistic if the condition was 1<=x<=75?
    PS. I am not mad 🙂

    • Thanks for writing! YES – you are definitely correct – I will fix this. I will also put in a note that the domain (number of t-shirts) must be an integer, but this shouldn’t affect the output of the problem. THANKS! Can you find more problems? haha Lisa

  13. Your website is so useful! But I would like to know how is it if the function has two absolute value? For example : g(x) = 3 |x-2| – |x+1|

    • Great question! I’d probably do use a number line to separate out the absolute values (like I did here), so we’d have g(x) = 3(2-x) – (-x-1), when x less than -1, g(x) = 3(2-x) – (x+1), when -1 less than or equal to x less than 2, and g(x) = 3(x-2) – (x+1), when x is greater than or equal to 2. Then I can simplify and put this into a piecewise function. Does that make sense? lisa

  14. Lisa what you’ve done here is amazing, I so much love the way you explain… I am however bugged by this assignment on limits of piecewise functions involving two absolute values.
    f(x) = {(2+/x/)÷3;x≥0
    {(/x/+4)÷3;x≤0
    Find: lim f(x)
    x→0.
    (Forgive my typo errors)

    • Thanks for writing and thanks so much for the kind words! For this one, I’d say the limit as x goes to 0 does not exist. This is because, from the left the graph gets closer and closer to 4/3 (and f(0) = 4/3) and from the right, the graph gets closer and closer to 2/3. Since the limit from the left does not equal the limit from the right, there is no limit as x -> 0. Does that make sense? Lisa

  15. I really like your website, it has helped me a lot!!

    Can you help me with the following problems?
    On the interval −3 ≤ x ≤ −1, what is a formula for f(x)?
    On the interval −1 ≤ x ≤ 1, what is a formula for f(x)?

  16. Your site is a wonderful resource! It is evident that you have put in a lot of time and hard work. Thank you so much for sharing with the rest of us!

  17. Hello, I’m in Algebra 1, 8th grade, and I have some questions regarding piecewise functions. What are the rules when writing piecewise functions? Also, how can you find the domain and range while looking at a graph and how do you put the domain and range into inequality format?

    Thank you for your time,
    Rebecca

  18. Your example problem post was so helpful. Most examples I come across and even the examples provided in the online math course I am taking do not explain as well as you did here. Often the steps are shown, but the reasoning and thought process is left out, but you did a great job giving full explanations and reasons behind the steps. Thank you! I will look to this site if I ever need help down the road.

    • Thanks so so much for writing and for your nice comments! Comments like yours make me want to keep writing Please spread the word about She Loves Math 😉 Lisa

  19. Hi, can you help me out with this problem? I can’t seem to solve it just be referring to your example above. Thanks!

    Nationwide Communications offers its customers a monthly plan for text messaging. Customers under this plan pay a flat rate of $5.00 each month for up to 200 text messages. For each additional text message over 200 (up to 1500 messages), the customer will pay $0.10. For each message over 1500 messages, the customer will pay $0.50.

    Complete the piecewise function that can be used to calculate the monthly billing amount, B, for a monthly level of t text messages. Note: Use the letter t as your variable.

    • Thanks for writing. Here’s how I’d do this problem: f(t) = 5, if t less than or equal to 200; 5 + .1(t – 200), if 201 less than or equal to t less than or equal to 1500; 5 + .1(1500 – 200) + .5(t – 1500), if t greater than or equal to 1501. (sorry – my less than and equal to signs don’t work in comments for some reason, so I wrote it out). Does that make sense? Lisa

  20. hi Lisa! Thank you so much for creating this site. I find it very helpful especially for those who have difficulties solving math problems. Keep it up! May you continue to be a blessing for those math lovers (and haters) : )

    -Ling, from the Philippines

  21. Your problem: f(x) = |X^2 -4| I think this one needs a bit more of a detailed explanation.
    Re: 2X + |X + 2| I think a little more explanation would be helpful.
    Your site is very enjoyable as a whole.

    • Thanks so much for writing, and I’ll work on making these problems more detailed. Let me know if you have any more feedback. Thanks again, Lisa

  22. For the second problem about t shirts would u have to add the totals from the previous costs before like the first choice u have or would u just have to leave it for that category like the second choice

    • Thanks for writing! I’m not exactly sure what you mean; the second column is the first column, but simplified – same result. When you say “choice”, do you mean column? Sorry I wasn’t more clear – maybe you could give me more information? Thanks, Lisa

  23. You have a summer job that pays time and a half for over time. That is, if you wprk more than 40 hours. Per week, you hourly wage for extra hours is 1.5 timesypir normal hpurly wage of $7.
    A. Write a piecewise function tha gives your weekly pay P in terms of the hours you work.
    B. How much will you get paid if you work 45?

  24. hi! good day! how do we know if the piece-wise function is continuous or not? and how to determine if the problem is a function or not?

    thank you very much! your reply is very much appreciated.

    • A function is continuous when, if you were to draw it, you never lift your pencil off the paper. So for a piecewise function to be continuous, where it changes from one function to another, the points have to touch, if that makes sense. The problem is a function when you never have more than one y for a given x. Does that help? Lisa

  25. can i ask something its our assignment and my our teacher dis not teach us

    if taxable income is 10,000 then the tas due is 5%

    over 10,000 but not over 30,000

    500+10% off the excess over 10,000

    ~~

    over 30,000 but not over 70,000

    2500+15% off the excess over 30,000

    ~~

    over 70,000 but not over 140,000

    8500+20% off the excess over 70,000

    ~~

    over 140,000 but not iver 250,000

    22,500+25% off the excess over 140,000

    ~~

    250,000 but not over 500,00

    50,000+30% off the excess over 250,000

    ~~

    over 500,000

    125,500+32% off the excess over 500

    ~

    income
    5,000
    10,000
    20,000
    50,000
    100,000
    150,000
    200,000
    300,000
    600,000

    please help me to answer or explain to me how to answer

  26. These examples were really helpful for me. Thank you very much.I was able to do my math seatwork independently because I looked at your examples. Keep doing what you’re doing because you can help many people like me. 🙆 You’re really great in Math, and I want to be great too. Huhu

  27. hi, i don’t understand this. I was tryna figure out how’d ya get the x and how you get the y and the graph thingy…..

  28. The fee for hiring a guide to explore a cave is P700.00. A guide only take care of a maximum of 4 persons, and additional guide can be hired as needed. Represent the cost of hiring guides as piece wise function of the number of tourists who wish to explore the cave. Help please

    • Thanks for writing! Here’s how I’d do this problem: Let x = the number of tourists, so we have y = 700, if 0 < x <= 4, y = 1400, if 5 <= x <= 8, 2100, if 9 <= x <= 12, and so on. YOu can also write this using an integer function: y = 700([(x-1)/4] + 1), where [] is the integer function. Does that make sense? Lisa

  29. Hi good evening.
    We are grouped from our math teacher to do a report tommorow regarding piece wise function.
    We are told to have an examples of that, and explain it in the front of the class butbthe problems is I have not the confidence to report because our teacher said that we must give a unique and simple equations and conditions.
    I just want to have from this site an atleast 3 simple examples with solutions and explainations. Im very thankful for the reply.

  30. research the income tax rate for individual taxpayers in the philippines . U can download this from http://www.bir.gov ph
    required:
    1.Represent the amount of tax as a function of income. Let x be the income and t(x) as the tax. (Note:there are 7 brackets)
    2. Find the amount of tax to be paid for the ff. income
    A.9000
    B.25000
    C.50000
    D.120000
    E.200000
    F.350000
    G.550000

    3.graph the function
    Pls help me

  31. Can you help me with this problem? I’m a beginner and i’m not pretty sure if I’ve done it correctly.

    A zumba instructor charges according to the number of participants. If there are 15 participants or below, the instructor charges Php 500 for each participant per month. If the number of participants is between 15 and 30, he charges Php 400 for each participant per month. If there are 30 participants or more, he charges Php 350 for each participant per month.

    Write the piecewise function that describes what the instructor charges.
    Graph the function.

  32. I am really weak in math,.. May u please help me? I have to report 3examples of piecewise funtion later.. And need to elaborate it,.. Please?? Cant contented enough the examples above…
    3_examples… Thank u..

    Grade11 student here.

  33. Hi I don’t know what to do in my project about piecewise function can you help me?

    Research about tax rates in the Philippines (e.g if you earn certain income level, then your tax rate is such and such
    Based on the assisting structure, ask students should define a piecewise function that models this situation by giving
    A. The function’s equation
    B. A table showing some function values
    C. A graph
    Your students will use the piecewise function, representing the tax structure or basic for discussing whether you think the tax rate is fair.

  34. Could you help me in this problem?
    Here is it.
    The fee for hiring a guide to explore a cave is 700. A guide can only take care of a maximum of 4 persons and additional guides can be hired as needed represent the cost of hiring guides as a function of the number of tourist who wish to explore the cave.

    • Here’s how I’d do this problem: y = 0, for x = 0. y = 700 * ([(x-1)/4] + 1) for x > 0, where x is an integer. Note that [ ] is an integer function, where you take the next lowest integer after the division. For example [1.5] = 1. Does that make sense?
      Lisa

      • SO if only 2 people show up then y=700[2/4] =0.??? Yet cost for 1,2,3,or 4 people would be 700. I read somewhere that the least integer is ]x[ so
        [1/4[=1?? So we jump up if fraction and stay if integer ]4/4[=1??

        • Thanks for writing back – you are absolute right! How about this: y = 0, for x = 0. y = 700 * ([(x-1)/4] + 1) for x > 0, where x is an integer. Lisa

    • Thanks so much for your nice comment – this means a lot. Keep visiting the site and let me know how I can make it better 😉 Lisa

    • I would write this as:

      f(x) = 40, if 0 less than or equal to x less than or equal to 500, 40 + ([(x – 500)/300] + 1)*3.5, if x > 500.

      Note that the [ ] function is the integer function, which means take the lower integer (round down) from what you get. Not sure if this is correct – does it look right?
      (for some reason, I had to spell out less than or equal to, or it wouldn’t work).
      Lisa

  35. Can someone help me with this problem?

    The cost c (in dollars) per unit of making x photocopies at The Office Shop is given below.

    18 cents for at most 20 copies
    12 cents for more than 20 copies up to 50
    9 cents for more than 50 copies but less than 100
    5 cents for 100 copies or more but up to 200
    3 cents for more than 200 copies

    Write the piecewise-defined function that represents the situation.

  36. how do you answer this question, an air conditioning salesperson receives a base salary of $2850 per month plus a commission. the commission is 2% of the sales up to and including $2500 for the month and 5% of the sales over $2500 for the month.

    • Thanks for writing! This could be a piece-wise function (and I think you meant 25000 instead of 2500?):
      y = 2850 + .02x; 0 is less than or equal to x is less than or equal to 25000
      y = 2850 + .05x; x is greater than 25000
      Does that make sense? Lisa
      (sorry, I had to spell out the inequalities since they didn’t show up right)

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