Inverse of Nonconstant Linear Functions

Inverses are very interesting creations. You can think of them as things that you use to "undo" what you just did. Say, let's take the variable $x$ and apply function $f$ to $x$ to get the result $f(x)$. Now, to get back to $x$ or "undo" are application of function $f$, we can take the inverse of $f$, which is written as $f$ to the $-1$ (don't confuse this as reciprocal). The inverse of $f$ would undo $f$ leaving $f^{-1}(f(x)) = x$. Yay, that was a wordy introduction. Here's a fun picture to understand this better.

Let's move onto some review of linear functions and how to find their inverses!

Linear functions are in the form $ax + by = c$ or more simply as $f(x) = y = mx + b$. So how do we find the inverses of these functions? Recall that for a function $f$, $f^{-1}(f(x)) = x$ (the inverse function $f^{-1}$ applied to the function $f$ gives the input $x$). Using this, we will be able to find the inverse of a linear function. Let's try an example out.

Example Find the inverse of $y = 3x + 5$.

Another way to write $y = 3x+5$ is the $f(x) = 3x+5$. Let us call $g(x)$ the inverse of $f(x)$. We know that $g(f(x)) = x$, meaning $g(x)$ must also be a linear function or else we will get powers of $x$ greater than $1$. Thus, we can assume $g(x) = ax+b$, where we have to find $a$ and $b$. Plugging $f(x)$ into our $g(x)$, we get $$g(f(x)) = a(3x+5)+b = x.$$ This means that $3ax + 5a+b = x$.

The linear terms on both sides of the equation must be the same, as do the constant terms on both sides of the equation. Namely, the coefficients must be the same. Therefore, $3a = 1$, so

$$a = \dfrac{1}{3}$$

and we know that $5a+b = 0$, so $$b = -\dfrac{5}{3}.$$ $f^{-1}(x)$, the inverse of $f$, or solely $g(x)$ must be $$g(x) = \dfrac{1}{3}x-\dfrac{5}{3}.$$ Checking our work, $g(f(x)) = \dfrac{1}{3}(3x+5) - \dfrac{5}{3} = x$, so we found the correct inverse! Continue for a better technique :D

Since we know that for every input $a$ passed into function $f$, we get output $b$ and for every output $b$ passed into $f^{-1}(x)$, the inverse of $f$, we will get back the input $a$. This means that what an inverse does, is switch the $x$ and $y$ of a function. Using our previous example, we can go ahead switch our $x$ and $y$ to get $x=3y+5$. Now solving for $y$, we get the same answer, $y = \dfrac{x-5}{3} = \dfrac{1}{3}x-\dfrac{5}{3}$!

From this trick of being able to switch $x$ and $y$ to find the inverse, we can see how the graphs of inverse functions behave.

We have that $g(x)=2x$ and when we calculate $g^{-1}(x)$, we find that $g^{-1}(x)=\frac{x}{2}$. And the graphs of $g(x)$ and its inverse are reflections of each other over the line $y=x$.

(Videos) Inverse Functions

by PatrickJMT

Here are some great video by PatrickJMT where he talks about what an inverse function is. An inverse function is a function that reverses what another function does. We denote functions with $f(x)$ and we denote inverse functions with $f^{-1}(x)$. It's very important to keep in mind that the $-1$ is NOT and exponent. It is simply just the way we denote inverse functions. Another important property is that not all functions will have inverses. You can use the horizontal line test to determine if the graph of a function will have an inverse. If there's any point where, if you draw a horizontal line, it touches two or more points, then the graph will NOT have an inverse.

The second video by PatrickJMT shows you how to find the inverse of a function. The steps he uses are:

1. Replace $f(x)$ with $y$.
2. Switch $x$'s and $y$'s.
3. Solve for $y$.
4. Replace $y$ with $f^{-1}(x)$.

(Video) Inverse Functions

by yaymath

Here's a video by yaymath showing you how to find the inverse of a function. An inverse function is just the reverse of a function. A function is denoted by $f(x)$ and an inverse function is denoted by $f^{-1}(x)$. It is important to note that the $-1$ is NOT and exponent but just a way to denote it's the inverse. To find the inverse function the steps are:

1. Flip the $x$'s and $y$'s in the equation.
2. Solve for $y$.

When looking at a graph, the inverse of a function is the function but flipped over the line $y=x$.

What is an Inverse of a Linear Function?

A function and its inverse undo each other. What exactly does that mean? Think of a function, often written like $f\left(x\right)$, like a math machine - you plug in one value and you get out another.

So, the inverse of a function is like a second machine that allows you to plug in the output of the first machine to get back to the original value.

The image below shows the relationship between a function and its inverse. Notice how we plug in $x$ to our function $f$ and get $y$. We can then plug in $y$ to our inverse function $f^{-1}$ and get back to $x$.

Consider this function and its inverse. Notice how we label the inverse by writing $f\left(x\right)^{-1}$. The letter used $\left(f,g,h...\right)$ doesn't matter, but it's most common to use $f$.

$$\begin{align} f(x) &= 4x - 5 \\ f^{-1}(x) &= \frac{x + 5}{4} \end{align}$$

Notice how, when we plug the output of the first function (3) into the inverse, we get back to the original input (2).

Let's try another example. This time we have

$$\begin{align} h(x) &= 1 + \sqrt{x - 2} \\ h^{-1}(x) &= (x - 1)^2 + 2 \end{align}$$

Notice again how, when we plug the output of the first function (1) into the inverse, we get back to the original input (2).

How Do You Find the Inverse of a Function?

You can actually figure out the inverse of a function by doing some simple rearranging of the original function.

Here are the steps to follow in order to find a function's inverse:

Let's practice this. Here's our original function:

$y=2x+5$

The solution looks like this:

$$\begin{align} x &= 2y + 5 \Rightarrow \\ x - 5 &= 2y \Rightarrow \\ \frac{x - 5}{2} &= y \end{align}$$

So, we've now figured out the inverse of our original function! It's $f^{-1}(x) = \frac{x - 5}{2}.$

(Video) How to Find the Inverse of a Function

by Nancy Pi

The general idea is that if a function $y = f(x)$ has an inverse, you can find it by writing the equation as $x = f(y)$ and solving for $y.$ Try it on this example where $f$ is linear!

(Video) Finding the Inverse of the Linear Function $f(x)=x+2$

by The Math Sorcerer

In this video by The Math Sorcerer an example of finding the inverse of a linear function is worked through.

Here is another example to try on your own.

Inverses occur in everyday life! Dirty dishes are washed to become new again. Borrowed library books are returned. The trip from school or work brings you back home.

Functions transform input values into output values.

Inverses transform the output values back into the input values they were before.

In this example: The input value is the test score.

The function is "Curved score = Test score + 10"

The output value is curved score.

Now: The input value is the curved score.

The function is "Test score = Curved score - 10"

The output value is the test score.

NOTATION

Now let's describe the example mathematically.

The first function would be: $y\left(x\right)=x+10$

The second function would be: $x\left(y\right)=y-10$

Because the second function transforms the outputs of the first function back into the inputs they were, the second function is the inverse of the first function.

Calculations are much easier and more organized if functions are put in terms of the independent variables. In this example, the curved score depends on the test score. Therefore, the test score is the independent variable.

So, how do we write the second function in terms of the independent variables and let people know it is the inverse of the first function?

We switch the variables in the first function: $x=y+10$

Then we solve for the dependent variable, y: $y=x-10$

Then we use a superscript of "-1" to mark the dependent variable as an inverse: $y^{-1}\left(x\right)=x-10$

In this case, this does NOT mean the inverse is a reciprocal of $y.$ Read the above function as "the inverse of $y$ of $x$ is $x$ minus $10$".

What if more than one input creates the same output?

Let's look at our diagram again.

In order for a function to be an inverse, it must undeniably change all of the outputs back into the original inputs. If more than one inputs create the same output, when the output goes into the inverse, we can not say for sure which input it came from.

Because we run into this problem, we would have to say that the function does not have an inverse or "the inverse does not exist". You can check if a function has an inverse by checking that no x-values will give you identical $y$-values. Graphically, you can use the horizontal line test.

The only kind of linear functions that do NOT have an inverse are constant functions in the form of $y=k$ where $k$ is a constant.

VIDEOS

Below are very useful videos about inverse functions. You will need them for the practice problems ahead.