Inverses of Functions

This section covers:

Note that there is an example of Transformation of Inverse functions here,  Inverses of Exponential and Log functions here, and Inverses of the Trigonometric Functions here.

What is the Inverse of a Function?

Getting the inverse of a function is simply switching the \(x\) and the \(y\), plotting the new graph (or doing the algebra to get the “new” \(y\)), and seeing what you get! Frequently, the inverse of a function won’t even be a function, since a function can’t have 2 “answers” \((y)\) for the same “question” \((x)\), but it can have 2 “questions” \((x)\) with the same “answer” \((y)\). Switching the \(x\) and \(y\) totally reverses these relationships and thus we may not always end up with another function; we may just have a “relation”.

We use the inverse notation \({{f}^{{-1}}}\left( x \right)\) to say we want the “normal” \((x)\) value back when \((y)\) is a certain number or expression. (Note that the notation has nothing to do with a negative exponent, which looks similar.)   

So for example, if the original function is \(f\left( x \right)=3x-4\), and we wanted \({{f}^{{-1}}}\left( 5 \right)\), we could get this by solving the equation \(5=3x-4\) to get  \(3\). In this case then, \({{f}^{{-1}}}\left( 5 \right)=3\). But later we’ll just switch the \((x)\) and \((y)\) variables and just solve for the new \((y)\) to get answers like this.


Inverses can be a little confusing, so let’s start out with an example.  Let’s say you and your foreign exchange student Justine (from France) are discussing the weather.  Justine says it’s 26 degrees Celsius outside and you want to convert that to degrees Fahrenheit. Then you say it’s supposed to be 90 degrees Fahrenheit for the weekend and Justine wants to know what that is in degrees Celsius

We actually saw these functions here in the Solving Algebraic Equations Section where we were solving one variable in terms of another.

Here are those functions and both their graphs on the same set of axes:

Let’s use this graph to answer the questions in the problem above:

“Justine says it’s 26 degrees Celsius outside and you want to convert that to degrees Fahrenheit.”  Since we want to know the temperature in degrees Fahrenheit, we look at the “F =” graph above.  We can see that when x is 26 degrees, y is about 79 degrees.

“It’s supposed to be 90 degrees Fahrenheit for the weekend and Justine wants to know what that is in degrees Celsius.”  Since we want to know the temperature in degrees Celsius, we look at the “C =” graph above.  We can see that when x is 90 degrees, y is about 32 degrees.  I’ve included these points on the graph.

Notice that the two functions (and, yes, they are both functions!) are symmetrical around the line “y = x”.  Symmetrical means that if you were to fold the piece of paper around that line, the two graphs would sit on top of each other; they are actually equidistant from the line.

This makes sense, since to get the inverse of a function, we are just switching the x and the y.

Finding Inverses Graphically

Since we can just switch the x and y to get the inverse of a function, we can easily do this with t-charts.  Here are some examples of functions and their inverses:

Let’s note a few things about the graphs above:

  • The domains and ranges in the original functions are reversed for the Inverses.  So the domain for the original function is the range for the inverse, and the range for the original function is the domain for the inverse.
  • For the \(y={{x}^{2}}\) function, in order to match up the inverse (square root) function, we had to also take the “minus” of the square root function; otherwise, we’d just have “half” the graph.  We’ll take about these special cases later.
  • In the first two examples above, the original function fails a Horizontal Line Test, meaning that you can draw a horizontal line somewhere across the function, and you’ll pass through more than one point.  Notice that the inverse relations then fail the Vertical Line Test, since we are just switching points.  Thus, the inverses in these cases are not functions.

Note that if a function has an inverse that is also a function (thus the original function passes the Horizontal Line Test, and the inverse passes the Vertical Line Test), the functions are called one-to-one.  This is because there is only one “answer” for each “question” for both the original function and the inverse function.

Finding Inverses Algebraically

If we are given the original function, finding the inverse of the function isn’t too bad.   The steps involved are:

  • Switch the \(x\)s and \(y\)’s in the equation.
  • Solve for the “new” \(y\).  When you get the “new” \(y\), replace it with \({{f}^{{-1}}}\left( x \right)\); this is inverse notation.   (Note that this has nothing to do with an exponent).
  • For original functions with even roots, we will have to add a restriction (where we have too much of the function when we get the inverse, so we have to restrict the domain.)
  • For original functions with even powers, we will have to include the other half of the inverse by using a plus/minus sign when we take a root.  Note that the inverse will not be a function in this case, since the original function doesn’t pass the horizontal line test (thus the inverse will not pass the vertical line test).

Note that a lot of times, to get the range of the original function, it’s easier to solve for the inverse, and see what that domain is (since the domains and ranges are switched for the inverse).  Remember again that we have to restrict the domain in a relation or function if (but not only if):

  • There is an x in a denominator, and that denominator could somehow be 0
  • There is an x inside an even radical sign and that radicand (inside the radical sign) could be negative
  • There is an indication anywhere in the problem that the domain is restricted

Here are some examples:

And here’s an example that is a one-to-one rational function, which is a function that has variables in the denominator of a fraction.   Don’t worry if you haven’t seen this; we’ll learn about these types of functions and asymptotes in the Graphing Rational  Functions, including Asymptotes section.  Note that the vertical asymptote of the original function is the horizontal asymptote of the inverses function, and vice versa.


Finding Inverses with Restricted Domains

Here are more advanced examples.  In the first example, we have a restricted domain when given the original; thus we have to restrict the range when we take the inverse.

In the second example, we are given a function, but asked to restrict the domain of the function so it is one-to-one, and then graph the inverse.  Remember that one-to-one means that both the original and the inverse are functions.

One way to think about these is that the original function must pass the Horizontal Line Test, so the inverse function can pass the Vertical Line Test.

Domains with Restrictions One-to-One

As you can see, sometimes it is easiest to get the domain and range by drawing the functions, and in both cases above, the original function passes the horizontal line test, and thus the inverse function passes the vertical line test.   For now, we can use t-charts to draw the graphs.

Don’t worry; it will be easier to draw “transformed” or “moved” functions when we learn about Parent Functions and Transformations.

Here are more examples where we want to find the inverse function, and domain and range of the original and inverse.

The second example is another rational function, and we’ll use a t-chart (or graphing calculator) to graph the original, restrict the domain, and then graph the inverse with the domain restriction:

More Inverses with Domain Restrictions

Using Compositions of Functions to Determine if Functions are Inverses

You can actually determine algebraically if two functions \(f\left( x \right)\) and \(g\left( x \right)\) are inverses of each other by using composition of functions:  if \(f\left( {g\left( x \right)} \right)=g\left( {f\left( x \right)} \right)=x\), then \(f\left( x \right)\) and \(g\left( x \right)\) are inverses.

Why?   Because you’re first plugging in an x to get out y, and then you plug in that y in the inverse, and out pops the original x again!  If you don’t get this, don’t worry; just remember that it works!

Let’s do some problems:

Determining if Inverses by Compositions of Functions

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 Parent Functions and Transformations – you are ready!

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