📌 What is an Inverse Function?

An inverse function reverses the operation of the original function. If a function takes \(x\) to \(y\), the inverse function takes \(y\) back to \(x\).

If \( f(x) = y \), then \( f^{-1}(y) = x \)

⚠️ Note: \( f^{-1} \) does NOT mean \( \frac{1}{f} \) (not the reciprocal!)

Requirement for Inverse:

A function must be one-to-one to have an inverse function.

Horizontal Line Test:

To determine if a function is one-to-one (and has an inverse), use the Horizontal Line Test:

Key Properties:

How to Use Tables:

To find the inverse function from a table, simply swap the input and output columns.

• The \(x\)-values of \(f\) become the \(y\)-values of \(f^{-1}\)
• The \(y\)-values of \(f\) become the \(x\)-values of \(f^{-1}\)

📝 Example:

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

Create table for \( f^{-1}(x) \) by swapping rows:

Examples from table:

• \( f^{-1}(5) = 1 \) because \( f(1) = 5 \)
• \( f^{-1}(9) = 3 \) because \( f(3) = 9 \)
• \( f^{-1}(11) = 4 \) because \( f(4) = 11 \)

How to Read Graphs:

To find \( f^{-1}(a) \) from the graph of \( f \):

1. Step 1: Find the point on the graph where the \(y\)-coordinate equals \(a\)
2. Step 2: Read the corresponding \(x\)-coordinate at that point
3. Step 3: That \(x\)-value is \( f^{-1}(a) \)

📝 Example:

Suppose the graph of \( f \) passes through these points:

\( (1, 3) \), \( (2, 7) \), \( (4, 15) \), \( (5, 19) \)

Find the following:

• \( f^{-1}(3) = 1 \) because \( f(1) = 3 \)
• \( f^{-1}(7) = 2 \) because \( f(2) = 7 \)
• \( f^{-1}(15) = 4 \) because \( f(4) = 15 \)
• \( f^{-1}(19) = 5 \) because \( f(5) = 19 \)

Reflection Property:

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

📝 Example - Graphing Inverse:

If \( f \) passes through points: \( (-2, 1) \), \( (0, 3) \), \( (2, 5) \), \( (4, 7) \)

Step 1: Swap coordinates for \( f^{-1} \)

\( f^{-1} \) passes through: \( (1, -2) \), \( (3, 0) \), \( (5, 2) \), \( (7, 4) \)

Step 2: Plot these points and connect

Step 3: Verify that the graphs are symmetric about \( y = x \)

Step-by-Step Process:

1. Step 1: Replace \( f(x) \) with \( y \)
2. Step 2: Swap \( x \) and \( y \) (interchange variables)
3. Step 3: Solve for \( y \) in terms of \( x \)
4. Step 4: Replace \( y \) with \( f^{-1}(x) \)
5. Step 5: Verify by checking composition

📝 Example 1 - Linear Function:

Find the inverse of \( f(x) = 3x - 7 \)

Step 1: Replace with \( y \)

\( y = 3x - 7 \)

Step 2: Swap \( x \) and \( y \)

\( x = 3y - 7 \)

Step 3: Solve for \( y \)

\( x + 7 = 3y \)
\( y = \frac{x + 7}{3} \)

Step 4: Write as \( f^{-1}(x) \)

\( f^{-1}(x) = \frac{x + 7}{3} \)

📝 Example 2 - Fraction:

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

Step 1: Replace with \( y \)

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

Step 2: Swap \( x \) and \( y \)

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

Step 3: Solve for \( y \)

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

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

📝 Example 3 - Square Root:

Find the inverse of \( f(x) = \sqrt{x - 2} + 3 \), where \( x \geq 2 \)

Steps:

\( y = \sqrt{x - 2} + 3 \)
\( x = \sqrt{y - 2} + 3 \)
\( x - 3 = \sqrt{y - 2} \)
\( (x - 3)^2 = y - 2 \)
\( y = (x - 3)^2 + 2 \)

\( f^{-1}(x) = (x - 3)^2 + 2 \), where \( x \geq 3 \)

Composition Method:

To verify that \( f \) and \( g \) are inverses, check BOTH compositions:

📝 Example - Verification:

Verify that \( f(x) = 2x - 5 \) and \( g(x) = \frac{x + 5}{2} \) are inverses

Check 1: Find \( f(g(x)) \)

\( f(g(x)) = f\left(\frac{x + 5}{2}\right) \)
\( = 2\left(\frac{x + 5}{2}\right) - 5 \)
\( = (x + 5) - 5 \)
\( = x \) ✓

Check 2: Find \( g(f(x)) \)

\( g(f(x)) = g(2x - 5) \)
\( = \frac{(2x - 5) + 5}{2} \)
\( = \frac{2x}{2} \)
\( = x \) ✓

Since both compositions equal \( x \), the functions ARE inverses!

Important Distinction:

📝 Example:

Consider \( f(x) = x^2 \) (with no domain restriction)

Original function: \( y = x^2 \)
Swap variables: \( x = y^2 \)
Solve for y: \( y = \pm\sqrt{x} \)

⚠️ This is NOT a function because for each positive \(x\), there are TWO values of \(y\). It's an inverse relation, but not an inverse function.

⚡ Quick Summary

• Inverse function reverses the original: \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \)
• Only one-to-one functions have inverse functions
• Horizontal Line Test: If any horizontal line crosses more than once, NO inverse function
• Point \( (a, b) \) on \( f \) → Point \( (b, a) \) on \( f^{-1} \)
• Domain of \( f^{-1} \) = Range of \( f \); Range of \( f^{-1} \) = Domain of \( f \)
• From tables: swap input and output columns
• From graphs: reflect across line \( y = x \)
• Finding algebraically: Replace with \( y \), swap \( x \) and \( y \), solve for \( y \)
• Verify: Check both \( f(g(x)) = x \) and \( g(f(x)) = x \)
• \( f^{-1} \) is notation, NOT \( \frac{1}{f} \)

📚 Key Formulas Reference