AP Precalculus: Inverse Functions

Master inverse functions with algebraic methods, graphical techniques, and table interpretations

🔄 5 Core Concepts 📐 Step-by-Step Methods 📊 Tables & Graphs 🎯 AP Exam Ready

📚 Understanding Inverse Functions

Inverse functions are a fundamental concept in AP Precalculus that "undo" the work of original functions. This guide covers how to identify, find, and work with inverse functions using algebraic, tabular, and graphical methods. Mastering these skills is essential for success on the AP exam and in advanced mathematics.

1 What is an Inverse Function?

An inverse function \(f^{-1}(x)\) reverses the action of \(f(x)\). If \(f\) takes input \(a\) and produces output \(b\), then \(f^{-1}\) takes input \(b\) and returns \(a\).

Inverse Function Identity \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\)
The composition of a function with its inverse equals the identity function

Essential Properties of Inverse Functions

Only one-to-one functions (functions that pass the horizontal line test) have inverse functions
The domain of \(f\) becomes the range of \(f^{-1}\) and vice versa
The graph of \(f^{-1}\) is a reflection of \(f\) over the line \(y = x\)
If \(f(a) = b\), then \(f^{-1}(b) = a\) — input and output are swapped

⚠️ Important Notation Warning

\(f^{-1}(x)\) is NOT the same as \(\frac{1}{f(x)}\). The superscript "-1" denotes the inverse function, not a reciprocal. For example, \(\sin^{-1}(x) \neq \frac{1}{\sin(x)}\).

📌 Conceptual Example

If \(f\) is a function that doubles a number and adds 3:

\(f(x) = 2x + 3\)

Then \(f^{-1}\) must undo this: subtract 3, then divide by 2:

\(f^{-1}(x) = \frac{x - 3}{2}\)

Verification: \(f(f^{-1}(5)) = f(1) = 2(1) + 3 = 5\) ✓

2 Finding Inverse Functions Algebraically

To find the inverse function algebraically, you essentially swap the roles of \(x\) and \(y\) and solve for the new \(y\). This reverses the input-output relationship.

Step-by-Step Method

Replace \(f(x)\) with \(y\) — Write the function as \(y = f(x)\)
Swap \(x\) and \(y\) — Interchange the variables to get \(x = f(y)\)
Solve for \(y\) — Isolate \(y\) on one side of the equation
Write the inverse — Replace \(y\) with \(f^{-1}(x)\)
Verify (optional) — Check that \(f(f^{-1}(x)) = x\)

📌 Example 1: Linear Function

Find the inverse of: \(f(x) = 2x + 3\)

Step 1: \(y = 2x + 3\)

Step 2: \(x = 2y + 3\) (swap \(x\) and \(y\))

Step 3: \(x - 3 = 2y\) → \(y = \frac{x - 3}{2}\)

Step 4: \(f^{-1}(x) = \frac{x - 3}{2}\)

📌 Example 2: Rational Function

Find the inverse of: \(f(x) = \frac{3x + 1}{x - 2}\)

Step 1: \(y = \frac{3x + 1}{x - 2}\)

Step 2: \(x = \frac{3y + 1}{y - 2}\)

Step 3: \(x(y - 2) = 3y + 1\) → \(xy - 2x = 3y + 1\) → \(xy - 3y = 2x + 1\) → \(y(x - 3) = 2x + 1\) → \(y = \frac{2x + 1}{x - 3}\)

Step 4: \(f^{-1}(x) = \frac{2x + 1}{x - 3}\)

📌 Example 3: Cube Root Function

Find the inverse of: \(f(x) = x^3 - 5\)

Step 1: \(y = x^3 - 5\)

Step 2: \(x = y^3 - 5\)

Step 3: \(x + 5 = y^3\) → \(y = \sqrt[3]{x + 5}\)

Step 4: \(f^{-1}(x) = \sqrt[3]{x + 5}\)

💡 Pro Tip

Always check your answer by verifying \(f(f^{-1}(x)) = x\). Substitute your inverse into the original function and simplify — you should get \(x\).

3 Finding Inverse Values from Tables

When given a table of values for a function \(f\), you can find values of the inverse function \(f^{-1}\) by reversing the input-output pairs. The key insight is: if \(f(a) = b\), then \(f^{-1}(b) = a\).

How to Read Inverse Values from a Table

To find \(f^{-1}(b)\), locate \(b\) in the output column (usually labeled \(f(x)\) or \(y\))
The corresponding value in the input column (usually labeled \(x\)) is \(f^{-1}(b)\)
Essentially, you're searching the \(f(x)\) column and returning the matching \(x\) value

📌 Example: Reading Inverse Values

Given table for \(f(x)\):

\(x\)	\(f(x)\)
1	5
2	6
3	8
4	9
5	12

Find these inverse values:

\(f^{-1}(5) = 1\) — because \(f(1) = 5\)

\(f^{-1}(6) = 2\) — because \(f(2) = 6\)

\(f^{-1}(9) = 4\) — because \(f(4) = 9\)

\(f^{-1}(12) = 5\) — because \(f(5) = 12\)

🔄 Creating an Inverse Table

To create the full inverse function table, simply swap the columns:
The \(f(x)\) column becomes the new \(x\) column, and the original \(x\) column becomes the new \(f^{-1}(x)\) column.

4 Finding Inverse Functions from Graphs

The graph of an inverse function \(f^{-1}\) is the reflection of the graph of \(f\) over the line \(y = x\). This means every point \((a, b)\) on \(f\) becomes \((b, a)\) on \(f^{-1}\).

Graphical Relationship If \((a, b)\) is on the graph of \(f\), then \((b, a)\) is on the graph of \(f^{-1}\)

Key Graphical Concepts

The line \(y = x\) acts as a "mirror" between \(f\) and \(f^{-1}\)
Intercepts swap: The \(x\)-intercept of \(f\) becomes the \(y\)-intercept of \(f^{-1}\)
Points where \(f\) crosses \(y = x\) are fixed points — they appear on both graphs
If \(f\) is increasing, \(f^{-1}\) is also increasing; same for decreasing

📌 Example: Reading Inverse Values from a Graph

If the graph of \(f\) passes through the points:

• \((2, 7)\) → then \(f^{-1}(7) = 2\)

• \((-1, 4)\) → then \(f^{-1}(4) = -1\)

• \((0, 3)\) → then \(f^{-1}(3) = 0\) (this is the \(y\)-intercept becoming an \(x\)-intercept)

📊 Horizontal Line Test

A function has an inverse if and only if it passes the Horizontal Line Test:
No horizontal line intersects the graph more than once.

💡 Sketching Inverse Graphs

To sketch \(f^{-1}\): (1) Plot the line \(y = x\), (2) Take key points from \(f\) and swap their coordinates, (3) Connect the new points maintaining the same curve shape reflected over \(y = x\).

5 Inverse Functions vs. Relations

While every relation has an inverse relation (created by swapping all input-output pairs), the inverse is only a function if the original was one-to-one. Non-one-to-one functions require domain restrictions.

When Inverses Aren't Functions

If \(f\) fails the horizontal line test, its inverse will fail the vertical line test
Example: \(f(x) = x^2\) (all real \(x\)) — the inverse \(y = \pm\sqrt{x}\) is not a function
Solution: Restrict the domain to make \(f\) one-to-one, then find the inverse

📌 Example: Domain Restriction

Function: \(f(x) = x^2\) for all real numbers

Problem: Not one-to-one (e.g., \(f(2) = f(-2) = 4\))

Solution 1: Restrict to \(x \geq 0\)

Inverse: \(f^{-1}(x) = \sqrt{x}\) with domain \(x \geq 0\)

Solution 2: Restrict to \(x \leq 0\)

Inverse: \(f^{-1}(x) = -\sqrt{x}\) with domain \(x \geq 0\)

⚠️ Common AP Exam Question

Be prepared to determine if an inverse exists, and if not, suggest an appropriate domain restriction that makes the function one-to-one.

📋 Inverse Function Properties Summary

🔤 Notation

\(f^{-1}\) means inverse of \(f\), NOT the reciprocal \(\frac{1}{f}\)

🔄 Composition Identity

\(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\)

📊 Graph Relationship

\(f^{-1}\) is the reflection of \(f\) over the line \(y = x\)

↔️ Domain-Range Swap

Domain of \(f\) = Range of \(f^{-1}\)
Range of \(f\) = Domain of \(f^{-1}\)

📐 Point Swap

If \((a, b)\) is on \(f\), then \((b, a)\) is on \(f^{-1}\)

✅ Existence Test

\(f^{-1}\) exists as a function ⟺ \(f\) is one-to-one (passes horizontal line test)