This section covers:
- Algebraic Functions Versus Relations
- Vertical Line Test
- Domain and Range of Relations and Functions
- Finding the Domain Algebraically
- More Practice
Algebraic Functions Versus Relations
When we first talked about the coordinate system, we worked with the graph that shows the relationship between how many hours we worked (the independent variable, or the “x”), and how much money we made (the dependent variable, or the “y”). Any relationship between two variables, where one depends on the other, is called a relation, since it relates two things.
This particular relation is an algebraic function, since there is only one y for each x. In other words, since the x is the “question” and y is the “answer”, we can only have one answer for each question. So for whatever is the number of hours we work, we only get paid a certain amount for that:
Again, a function is just a fancy way of saying something depends on something else, and there’s only one “y” for every “x”. But the other thing you’ll learn about functions is that they can be written a funny way; a way that looks really complicated, but they are just trying to confuse us – it’s not that bad!
Instead of our original equation, y = 10x, we can write it like this:
Note that this is not “f times x”; it is “f of x”. What it means is that x is on the right hand sign of the “=” sign, and you can put different values in for x on the left hand side to get one and only one value on the right hand sign. So again, “f(x)” is really “y”. It’s that simple.
Again, what makes a relation a function is that you can only have one “answer” (the y) for each “question” (the x). So all functions are relations, but not all relations are functions.
Vertical Line Test
Notice that when we have a function, we can’t draw a vertical that goes through more than one point. This is called the vertical line test, and it’s a useful tool to determine if a graph is a function or not. For example, we can tell the following graph is not a function since we can draw a vertical line and hit more than one point:
Here are some examples of functions. Notice that you can have two “questions” (x) for the same “answer” (y):
Another way to look at a set of points and determine whether or not they are functions is to draw what we call mapping diagrams, since we are mapping the x values to the y values. We order values from smallest to largest and don’t repeat the values on each side and match them up. If we have more than one y value for one x value, we don’t have a function. Here are some examples:
Domain and Range of Relations and Functions
Domain and Range of functions (and relations) sound really difficult and scary, but they are not really bad at all. You know how those mathematicians like to use fancy words for easy stuff?
Remember that since “d” comes before “r”, the domain of functions have to do with the “x”’s and the range of functions have to do with the “y”’s. To get the domain, we are just looking for all the possible values of x for that function (from smallest to largest), and for the range, we are looking for all possible values of y for that function (again, from smallest to largest).
To help me do this, I like to use my pencil – but it’s backwards compared to what you might think. To find the domain, I put my pencil vertically and start at the left and see where it first hits a point. Then I push it through all the way to the right to see where it ends hitting points.
For the range, I do the same thing, but with a horizontal pencil that’s moving up:
Here are more examples, using what we call “Interval Notation”. (We saw this in the Inequalities Section). This is the most commonly used way to describe domains and ranges, and it always goes from lowest to highest with “(“ (soft brackets) if the relation or function doesn’t hit the point, and “[" (hard brackets) if the relation or function does hit the line. If you have to skip over any numbers, you do so by using the "U” sign, which means union, or putting things together.
We can also use Inequality Notation, where, as we saw before, we use inequality signs to describe the ranges.
Notice on when we see arrows (in the first two examples below), we have to assume that both the domain and range go on forever in those directions.
If you don’t see how we got the domain and range above, use the pencil trick, and make sure you start from the left for the domain (with vertical pencil) and from the bottom with the range (with horizontal pencil).
Note again that the last two graphs are not functions; they do not meet the Vertical Line Test requirements.
Restricted Domains: Finding the Domain Algebraically
In many cases, the domain is restricted.
A domain is restricted if: it is indicated that way in the problem, and/or there is a variable in the denominator and the denominator could be zero and/or there is a variable underneath an even radical sign and that radicand could be less than zero.
It could also be restricted if there are other functions in it, like the log function, but we’ll learn about that later in the Logarithmic Functions section.
We start out assuming that the domain of a function is all real numbers, but then see if there are any exceptions, as seen in the table. We will learn more about rational functions (shown in the first two examples, where there are variables in the denominator) in the Rational Functions and Equations, and Graphing Rational Functions, including Asymptotes sections.
We will work on more advanced topics with functions later, in the Advanced Functions section.
Learn these rules, and practice, practice, practice!
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On to Scatter Plots, Correlation, and Regression – you are ready!