Inverse Functions

We can think of a function as a transformation. In this lesson, we will look at whether or not we can reverse that transformation. This will give you another way to build new functions out of some already-known functions, by taking their inverses.


Definition of inverse functions

The function $\cl"red"{f}$ from the previous lesson is given in a table below, along with a new function $\cl"blue"{g}$. They are also graphed on the grids to the left. Notice that the graphs of $\cl"red"{f}$ and $\cl"blue"{g}$ are the same shape, but with their $x$ and $y$ coordinates reversed. (This is the same as reflecting the graphs around the dashed line $y=x$.)

$x$$f(x)$
$y$$g(y)$

For each $x$ in the table below, compute ${g ∘ f}(x)$.

$x$${g ∘ f}(x)$
Is ${g ∘ f}(x)$ always equal to $x$?

You could also check that ${f ∘ g}(y)$ is always equal to $y$, in the same way. This means that $\cl"red"{f}$ and $\cl"blue"{g}$ undo each other. That is, when you use $\cl"red"{f}$ to transform a number into another number, $\cl"blue"{g}$ will transform the result back into the original number, and vice versa.

If $f$ and $g$ are two functions with ${g ∘ f}(x)=x$ for every $x$ in the domain of $f$, and ${f ∘ g}(y)=y$ for every $y$ in the domain of $g$, then we call $f$ and $g$ inverse functions.

One-to-one functions

For a function $f$ to have an inverse function, it must be possible to tell which value of $x$ is transformed into each value of $f(x)$.

The function $\cl"red"f$ is given in the table to the right, and also graphed on the grid to the left. How many different values of $x$ are there with $\cl"red"{f(x)=3}$?
If you know that $\cl"red"{f(x)=3}$, can you tell what $x$ is?
$x$$\cl"red"{f(x)}$
$-1$$1$
$1$$3$
$2$$3$
$3$$4$
$4$$2$
If $g$ was an inverse function to $\cl"red"f$, would it be possible to assign a single value to $g(3)$?
Does $\cl"red"f$ have an inverse function?

A function $f$ is called one-to-one if $f(x)$ has a different value for every distinct (different) value of $x$.

Is $\cl"red"f$ a one-to-one function?
A new function $\cl"red"h$ is given in the table to the right, and also graphed on the grid to the left. Is $\cl"red"h$ a one-to-one function?
$x$$\cl"red"{h(x)}$
$0$$2$
$1$$3$
$2$$-4$
$3$$1$
$4$$-2$

If $\cl"red"h$ has an inverse function, we’ll name that function $\cl"blue"{h^{-1}}$ (read as “$h$-inverse”). Click to show a grid where we can graph $\cl"blue"{h^{-1}}$.

Because $\cl"red"{h(0)=2}$, we know that $\cl"blue"{h^{-1}(2)=0}$ (since $\cl"blue"{h^{-1}}$ must reverse the effect of $\cl"red"h$). Proceeding in this way, fill in the table below for $\cl"blue"{h^{-1}}$. Your answers will be plotted on the bottom grid to the left.

$y$$\cl"blue"{h^{-1}(y)}$

When a function is one-to-one, we can always find an inverse of it in this way. In fact:

Functions have inverses precisely when they are one-to-one. If $f$ is a one-to-one function, we will write $f^{-1}$ for the inverse function of $f$.

What are the domain and range of $\cl"red"h$?

Domain:
Range:

What are the domain and range of $\cl"blue"{h^{-1}}$?

Domain:
Range:

If $f$ has an inverse, the domain of $f$ will be the range of $f^{-1}$, and the range of $f$ will be the domain of $f^{-1}$.

Inverses of some specific functions

Just as you can tell if a relation is a function by looking at its graph, you can also tell if a function is one-to-one by looking at its graph.

If a function has two or more inputs that give the same output, it is not one-to-one. Looking at the two functions shown below, the one on the right isn’t one-to-one because there are two different inputs that will give you each output greater than 0 (that is, two different points on the graph at each height greater than zero). The function on the left is one-to-one because each point on the graph is at a different height.

The function $\cl"red"f$ given by $\cl"red"{f(x)=x^3}$ is graphed on the grid to the left.

Based on the graph of $\cl"red"{f(x)}$, is $\cl"red"{f}$ a one-to-one function?
Does $\cl"red"{f}$ have an inverse?
Compute $\cl"red"{f(2)}$.
Using your previous answer, what is $f^{-1}(8)$? (Remember that $f^{-1}$ is the name given to the inverse function of $f$.)

If $\cl"red"{f(x)=x^3}$, then $f^{-1}(y)$ must be the number whose cube is $y$. When $y$ is positive, we have already defined that number to be $√^{3} y$. As you can see by looking at the graph, the range of the function $\cl"red"f$ is the set of all real numbers, so it makes sense to talk about $f^{-1}(y)$ for any real number $y$ (including negative numbers).

As a result, we will define $√^{3} y$, the cube root of $y$, to be the value $f^{-1}(y)$, where $\cl"red"{f(x)=x^3}$. This definition works whether $y$ is negative or positive.

Click to see the graph of $\cl"blue"{f^{-1}(y)=√^{3}y}$ below the graph of $\cl"red"f$. Notice that the two graphs are the same shape, but with the axes switched.

What is $\cl"red"{f(-3)}$?
What is $\cl"blue"{f^{-1}(-27)}$?
What is $\cl"red"{f(-4)}$?
What is $\cl"blue"{√^{3}{-64}}$?

The function $\cl"green"f$ given by $\cl"green"{f(x)=x^2}$ is graphed on the grid to the left.

Is $\cl"green"{f}$ one-to-one?
Does $\cl"green"{f}$ have an inverse?

Click to see the graph of a new function $\cl"red"{h}$. This is a restriction of $\cl"green"{f}$ to a smaller domain: that is, the domain of $\cl"red"{h}$ is smaller than the domain of $\cl"green"{f}$, but inside the domain of $\cl"red"{h}$ the functions have the same value at every point.

Is $\cl"red"{h}$ one-to-one?
Does $\cl"red"{h}$ have an inverse?

Because $\cl"red"{h(x)=x^2}$ whenever $x≥0$, $h^{-1}(x)$ must be the non-negative number whose square is $x$. That is, $h^{-1}(x)$ is the square root of $x$.

Click to see the graphs of $\cl"red"{h}$ and $\cl"blue"{h^{-1}}$.

Is $\cl"red"{h(x)}$ ever a negative number?
In the previous question, you saw that we could define the cube root of a negative number. Is it possible to define the square root of a negative number in the same way?

If $f$ is a linear function, you can compute $f^{-1}$ using algebra.

For example, the function $\cl"red"f$ given by $\cl"red"{f(x)=2x+1}$ is graphed on the top grid to the left, and its inverse is graphed on the bottom grid. We’d like to find a formula for $\cl"blue"{f^{-1}(y)}$. That is, given any $y$ value, our formula should produce the value of $x=f^{-1}(y)$ such that $f(x)=y$. To find this formula, we start with the equation $f(x)=y$, and then solve for $x$ in terms of $y$:

$$ \cl"tight"{\table 2x, +1, =, y; 2x, , =, y, -, 1; x, , =, \colspan 3{1/2(y-1)}; x, , =, {1/2}y, -, 1/2 } $$

So $$\cl"blue"{f^{-1}(y)={1/2}y - 1/2}$$.

Click to see the graph of $\cl"red"{g(x)=3x-4}$ and its inverse. Find a formula for $\cl"blue"{x=g^{-1}(y)}$, by solving $g(x)=y$ for $x$ in terms of $y$.

Now click to see the graph of $\cl"red"{h(x)=x-2}$ and its inverse.

Using scratch paper or the graph to the left, what is a formula for $\cl"blue"{h^{-1}}$?

To complete the table below, click the Next buttons to see the graphs of the linear functions $\cl"red"f$, $\cl"red"g$, and $\cl"red"h$ again, along with the graphs of their inverses. Enter the slope of each graph, and the formulas you just found for the inverse functions.

FunctionSlopeInverseSlope of
inverse

As you can see from this table:

If $f$ is a linear function with slope $m$ and $m ≠ 0$, then $f^{-1}$ is a linear function with slope $$1/m$$.

Click to see the graph of the function $\cl"red"k$ given by $\cl"red"{k(x)=-2x+4}$.

What is the slope of $k$?
What is the slope of $k^{-1}$?
By looking at this graph, find the intercepts of $\cl"red"{k}$. (The intercepts of a graph are the places where it crosses the $x$-axis and the $y$-axis.)

Now, click to see the graphs of $\cl"red"{k}$ and $\cl"blue"{k^{-1}}$.

Where are the intercepts of $\cl"blue"{k^{-1}}$?

Because $\cl"red"{k}$ and $\cl"blue"{k^{-1}}$ have the same shape except with the coordinates switched, they also have the same intercepts except with the coordinates switched. This is true of any linear function:

If $f$ is a linear function with intercepts $(0,b)$ and $(c,0)$, then $f^{-1}$ is a linear function with intercepts $(b,0)$ and $(0,c)$.