Verifying inverse functions by composition

▶ Verifying inverse functions from tables

Inverse functions undo each other. Functions s and t are inverses if and only if s(t(x))=x and t(s(x))=x for every x-value in the domains.

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▶ Using specific values to test for inverses

Are two functions inverses? Even one counterexample can show that they are not. How many examples would it take to show that they really are inverses?

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▶ Verifying inverse functions by composition

Sal composes f(x)=(x+7)³-1 and g(x)=∛(x+1)-7, and finds that f(g(x))=g(f(x))=x, which means the functions are inverses!

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▶ Verifying inverse functions by composition: not inverse

Sal composes f(x)=2x-3 and g(x)=½x+3, and finds that f(g(x)) ≠ g(f(x)) ≠ x, which means the functions are not inverses.

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📄 Verifying inverse functions by composition

This article includes a lot of function composition. If you need a review on this subject, we recommend that you go here before reading this article.

Inverse functions, in the most general sense, are functions that "reverse" each other. For example, if a function takes $a$ to $b$, then the inverse must take $b$ to $a$.

Let's take functions $f$ and $g$ for example: $f(x)=\dfrac{x+1}{3}$ and $g(x)=3x-1$.

Notice how $f(5)=2$ and $g(2)=5$. [[☃ explanation 1]]

Here we see that when we apply $f$ followed by $g$, we get the original input back. Written as a composition, this is $g(f(5))=5$.

But for two functions to be inverses, we have to show that this happens for all possible inputs regardless of the order in which $f$ and $g$ are applied. This gives rise to the inverse composition rule.

##The inverse composition rule

These are the conditions for two functions $f$ and $g$ to be inverses:

$f(g(x))=x$ for all $x$ in the domain of $g$
$g(f(x))=x$ for all $x$ in the domain of $f$

This is because if $f$ and $g$ are inverses, composing $f$ and $g$ (in either order) creates the function that for every input returns that input. We call this function “the identity function".

###Example 1: Functions $f$ and $g$ are inverses

Let's use the inverse composition rule to verify that $f$ and $g$ above are indeed inverse functions.

Recall that $f(x)=\dfrac{x+1}{3}$ and $g(x)=3x-1$.

Let's find $f(g(x))$ and $g(f(x))$.

$f(g(x))$	$\qquad \qquad g(f(x))$
$\begin{align} f(\greenD{g(x)})&=\dfrac{\greenD{g(x)}+1}{3}\\&=\dfrac{\greenD{3x-1}+1}{3}\\&=\dfrac{3x}{3}\\&=x\\end{align}$	$\qquad\qquad \begin{align}g(\purpleC{f(x)})&=3\left(\purpleC{f(x)}\right)-1\\&=3\left(\purpleC{\dfrac{x+1}{3}}\right)-1\\&=x+1-1\\&=x\\end{align}$

So we see that functions $f$ and $g$ are inverses because $f(g(x))=x$ and $g(f(x))=x$.

###Example 2: Functions $f$ and $g$ are not inverses

If $f(g(x))$ or $g(f(x))$ is not equal to $x$, then $f$ and $g$ cannot be inverses.

Let's try this for $f(x)=5x-7$ and $g(x)=\dfrac{x}{5}+7$.

$f(g(x))$	$\qquad g(f(x))$
$\begin{align} f(\greenD{g(x)})&=5(\greenD{g(x)})-7\\&=5\left(\greenD{\dfrac{x}{5}+7}\right)-7\\&=x+35-7\\&=x+28\end{align}\qquad $	$\qquad\begin{align} g(\purpleC{f(x)})&=\dfrac{\purpleC{f(x)}}{5}+7\\&=\dfrac{\purpleC{5x-7}}{5}+7\\&=x-\dfrac75+7\\&=x+\dfrac{28}{5}\\end{align}$

So functions $f$ and $g$ are not inverses because $f(g(x))\neq x$ and $g(f(x))\neq x$.

(Note here, that we could have concluded that $f$ and $g$ were not inverses after showing that $f(g(x))=x+28$.)

Check your understanding

In general, to check if $f$ and $g$ are inverse functions, we can compose them. If the result is $x$, the functions are inverses. Otherwise, they are not.

[[☃ graded-group 1]]

[[☃ explanation 1]]

[[☃ graded-group 1]]

[[☃ explanation 1]]

[[☃ graded-group 1]]

[[☃ explanation 1]]

📄 Composite and inverse functions: FAQ

Why do we need to know how to compose functions?

Composing functions allows us to combine two or more functions into one. We can use this technique to build more complicated functions from simpler ones. It also allows us to model more complex situations in math, physics, engineering, and other disciplines.

What is an invertible function?

An invertible function is one that can be "undone" or reversed. In other words, we can find another function that will take us back to where we started. For example, a function that adds $2$ to a number is invertible: the inverse function would subtract $2$ from a number.

##Why do we sometimes need to restrict the domain of a function to make it invertible?

In general, we don't always need to restrict the domain of a function to make it invertible. However, there are certain cases where it is necessary.

To be invertible, a function must have the property that each input maps to a unique output. Otherwise, the inverse function won't be able to "undo" the original function unambiguously.

For example, in the function $f(x) = x^2$, both $-2$ and $2$ map to $4$. Should the inverse function map $4$ back to $2$ or $-2$? It's ambiguous, but a function can only have one output for a given input.

However, if we restrict the domain of $f$ to $x≥ 0$ (non-negative numbers), then the function becomes invertible.

How do we graph inverse functions?

The easiest way to graph an inverse function is by reflecting the original function across the line $y=x$. This is because the inputs and outputs switch places when we invert a function.

##Where are composite and inverse functions used in the real world?

These functions are used in a lot of places! For example, in physics, we might use a composite function to model the motion of a falling object. In economics, we might use an inverse function to model the relationship between supply and demand.