Inverse Functions: Definitions, Properties, and Graphs

Inverse Functions

Definition of the Inverse of a Function

An inverse function essentially reverses the roles of the input and output of the original function. If a function f is defined by ordered pairs $(x, y)$, then its inverse, denoted as $f^{-1}$, consists of the ordered pairs $(y, x)$.

• Notation: The inverse of $f$ is written as $f^{-1}$ (read as "f-inverse").

• Key Property: If $f(a) = b$, then $f^{-1}(b) = a$.

Graphs of f and f Inverse

The graph of $f^{-1}$ is a reflection of the graph of $f$ about the line $y = x$. This means that every point $(a, b)$ on the graph of $f$ corresponds to a point $(b, a)$ on the graph of $f^{-1}$.

• Domain and Range: The domain of $f$ is the range of $f^{-1}$, and the range of $f$ is the domain of $f^{-1}$.

Example 1: Graphing an Inverse Function

Given the graph of $f$, you can draw the graph of its inverse by reflecting each point across the line $y = x$. State the domain and range of both $f$ and $f^{-1}$.

Horizontal Line Test for Inverse Functions

A function $f$ has an inverse that is also a function if and only if no horizontal line intersects the graph of $f$ more than once. This is known as the Horizontal Line Test.

• If $f$ passes the horizontal line test, it is called a one-to-one function.

• Only one-to-one functions have inverses that are also functions.

Verifying Inverses

To verify that two functions $f$ and $g$ are inverses of each other, you must show:

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

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

If both conditions are satisfied, then $f$ and $g$ are inverses of each other.

Finding the Inverse of a Function Algebraically

To find the inverse of a function $f(x)$ algebraically, follow these steps:

1. Replace $f(x)$ with $y$ in the equation for $f(x)$.

2. Interchange $x$ and $y$.

3. Solve for $y$. Replace $y$ with $f^{-1}(x)$.

4. Verify your result by showing that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

Examples of Finding Inverses

• Example 3: Show that $f(x) = 4x - 7$ and $g(x) = \frac{x + 7}{4}$ are inverses.

  • Compute $f(g(x)) = 4\left(\frac{x + 7}{4}\right) - 7 = x + 7 - 7 = x$

  • Compute $g(f(x)) = \frac{4x - 7 + 7}{4} = \frac{4x}{4} = x$

  • Thus, $f$ and $g$ are inverses.

• Example 4: Find the inverse of $f(x) = 2x + 7$.

  • Let $y = 2x + 7$

  • Switch $x$ and $y$: $x = 2y + 7$

  • Solve for $y$: $x - 7 = 2y \implies y = \frac{x - 7}{2}$

  • So, $f^{-1}(x) = \frac{x - 7}{2}$

• Example 5: Find the inverse of $f(x) = 3x + 4$.

  • Let $y = 3x + 4$

  • Switch $x$ and $y$: $x = 3y + 4$

  • Solve for $y$: $x - 4 = 3y \implies y = \frac{x - 4}{3}$

  • So, $f^{-1}(x) = \frac{x - 4}{3}$

Graphing Inverses

To graph $f$ and $f^{-1}$ on the same axes, plot the original function and then reflect each point across the line $y = x$ to obtain the graph of the inverse. State the domain and range of both $f$ and $f^{-1}$.

Additional info: For piecewise or restricted domain functions, the inverse may only exist on certain intervals. Always check the domain and range when finding and graphing inverses.