Inverse Functions and Their Properties in Precalculus

Inverse Functions

Definition and Properties

An inverse function reverses the effect of the original function. If $f$ is a function, its inverse $f^{-1}$ satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ for all $x$ in the domain of $f$ and $f^{-1}$ respectively.

• One-to-One Function: A function is one-to-one (injective) if each output is produced by exactly one input. This is necessary for a function to have an inverse.

• Horizontal Line Test: A function is one-to-one if and only if no horizontal line intersects its graph more than once.

• Invertibility: Only one-to-one functions are invertible. If a function fails the horizontal line test, it does not have an inverse.

Example: The function $f(x) = 2x + 3$ is one-to-one and invertible. The function $f(x) = x^2$ (on $(-c,5c)$) is not one-to-one and not invertible.

Justifying Invertibility

• If a function passes the horizontal line test, it is invertible.

• If a function fails the horizontal line test, it is not invertible.

Example: Given a graph, if every horizontal line crosses the graph at most once, the function is invertible.

Finding the Inverse of a Function

Step-by-Step Process

1. Replace $f(x)$ with $y$.

2. Interchange $x$ and $y$.

3. Solve for $y$.

4. Replace $y$ with $f^{-1}(x)$.

Example 1: Find the inverse of $f(x) = 3x - 4$.

• Step 1: $y = 3x - 4$

• Step 2: $x = 3y - 4$

• Step 3: $x + 4 = 3y ightarrow y = \frac{x + 4}{3}$

• Step 4: $f^{-1}(x) = \frac{x + 4}{3}$

Example 2: Find the inverse of $f(x) = \frac{x + 1}{2}$.

• Step 1: $y = \frac{x + 1}{2}$

• Step 2: $x = \frac{y + 1}{2}$

• Step 3: $2x = y + 1 \rightarrow y = 2x - 1$

• Step 4: $f^{-1}(x) = 2x - 1$

Verifying Inverse Functions

Algebraic Verification

To verify that $f$ and $g$ are inverses, check:

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

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

Example: Let $f(x) = 3x + 5$, $g(x) = \frac{x - 5}{3}$.

• $f(g(x)) = 3\left(\frac{x - 5}{3}\right) + 5 = x - 5 + 5 = x$

• $g(f(x)) = \frac{3x + 5 - 5}{3} = \frac{3x}{3} = x$

Therefore, $f$ and $g$ are inverse functions.

Domain and Range of Inverse Functions

Relationship Between Domain and Range

• The domain of $f$ becomes the range of $f^{-1}$.

• The range of $f$ becomes the domain of $f^{-1}$.

Example: If $f$ has domain $[2, 4]$ and range $[-2, 3]$, then $f^{-1}$ has domain $[-2, 3]$ and range $[2, 4]$.

Tables and Mapping Diagrams

Using Tables to Find Inverse Values

Given a table of values for $f$, the inverse $f^{-1}$ maps outputs back to their corresponding inputs.

x	f(x)
1	6
2	8
3	4
4	5
5	1

• Example: $f^{-1}(5) = 4$ because $f(4) = 5$.

• Example: To solve $f^{-1}(3) = ?$, find $x$ such that $f(x) = 3$.

Graphing Inverse Functions

Graphical Relationship

• The graph of $f^{-1}$ is the reflection of the graph of $f$ across the line $y = x$.

• To sketch $f^{-1}$, swap the $x$ and $y$ coordinates of each point on $f$.

Example: If $f$ passes through $(2, 5)$, then $f^{-1}$ passes through $(5, 2)$.

Summary Table: Properties of Invertible Functions

Property	Function $f$	Inverse $f^{-1}$
Domain	$D_f$	$R_f$
Range	$R_f$	$D_f$
One-to-One	Required	Required
Graph	Original	Reflection over $y = x$

Worked Examples

Example 1: Find the Inverse

• Given $f(x) = 5x + 2$, find $f^{-1}(x)$.

• Step 1: $y = 5x + 2$

• Step 2: $x = 5y + 2$

• Step 3: $x - 2 = 5y \rightarrow y = \frac{x - 2}{5}$

• Step 4: $f^{-1}(x) = \frac{x - 2}{5}$

Example 2: Verify Inverse Functions

• Let $f(x) = 3x - 4$, $f^{-1}(x) = \frac{x + 4}{3}$.

• $f(f^{-1}(x)) = 3\left(\frac{x + 4}{3}\right) - 4 = x + 4 - 4 = x$

• $f^{-1}(f(x)) = \frac{3x - 4 + 4}{3} = \frac{3x}{3} = x$

Key Takeaways

• Only one-to-one functions are invertible.

• The inverse function undoes the action of the original function.

• To find the inverse, swap $x$ and $y$ and solve for $y$.

• The graph of the inverse is a reflection over $y = x$.

• Domain and range swap between a function and its inverse.