Section 4.1 - Inverse Functions

Section Objectives

1. Identify one-to-one functions.
2. Determine whether two functions are inverses.
3. Find the inverse of a function.

One-to-One Functions

A function is one-to-one if different input values always produce different output values. In other words, a function is one-to-one if no $y$-values are reused.

Examples

• $f(x)=x^2$ is not one-to-one because $f(2)=f(-2)=4$. The value $y=4$ is associated with two different $x$-values, $2$ and $-2$.

• $g(x)=\sqrt{x}$ is one-to-one because no two different numbers have the same square root.

A function is one-to-one if and only if its graph passes the horizontal line test. (Remember that the graph of any function must pass the vertical line test.)

• Let $f(x)=3x+5$. Is $f$ a one-to-one function?

• Is $f(x)=1/x$ one-to-one? What about $g(x)=1/x^2$?

Inverse Functions

The inverse of a function is a new function that undoes the action of the original function.

Examples

• $f(x)=2x+1$ and $\displaystyle g(x)=\frac{x-1}{2}$ are inverses. What one does, the other undoes.
• Not all functions have inverse functions. For example, the constant function $f(x)=1$ does not have an inverse.
• One-to-one functions have inverses.

Formal definition of Inverse: The functions $f$ and $g$ are inverses if $f(g(x))=x$ for all $x$ in the domain of $g$ and $g(f(x))=x$ for all $x$ in the domain of $f$.

When $g$ is the inverse of $f$, we write $g=f^{-1}$.

If $f^{-1}$ exists, then

• $f$ and $f^{-1}$ are one-to-one functions,
• the domain of $f^{-1}$ is the range of $f$,
• the range of $f^{-1}$ is the domain of $f$,
• $f^{-1}(f(x))=x$ for all $x$ in the domain of $f$,
• $f(f^{-1}(x))=x$ for all $x$ in the domain of $f^{-1}$.

The last two bullet points above reiterate the fact that inverse functions undo the action of each other: $y=f(x)$ if and only if $f^{-1}(y)=x$.

VERY IMPORTANT IDEA: If $(x,y)$ is a point on the graph of $f$, then $(y,x)$ is a point on the graph of $f^{-1}$. Since $(y,x)$ is the reflection of $(x,y)$ about the line $y=x$, it follows that the graph of $f^{-1}$ can be obtained from that of $f$ by reflecting it about the line $y=x$.

Examples

• Show that $f(x)=(x-2)^3+1$ and $g(x)=\sqrt[3]{x-1}+2$ are inverses.

• Let $f(x)=6x+5$. Compute $f^{-1}(29)$.

• Find the inverse of the function defined by $\{(1,2), (2,4), (3,6), (4,8), (5,10)\}$.

Finding Inverses

For some functions, we can use algebra to determine inverse functions. Essentially, to find an inverse function we "undo" the original function.

Given $y=f(x)$,

1. Find the domain of $f$ and verify that $f$ is one-to-one.
2. Solve for $x$ in terms of $y$.
3. Interchange $x$ and $y$ to get $y=f^{-1}(x)$.
4. Define the domain of $f^{-1}$ to be the range of $f$.

Examples

• Find the inverse of $f(x)=2x+1$.

• Find the inverse of $g(x)=\sqrt[3]{x-5}$.