Inverse Functions
Complete Notes & Formulae for Twelfth Grade (Precalculus)
1. Identify Inverse Functions
Definition of Inverse Function:
A function \( f^{-1} \) is the inverse of function \( f \) if:
\[ f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x \]
In simpler terms: The inverse function "undoes" what the original function does
Key Property:
• If \( (a, b) \) is on the graph of \( f \), then \( (b, a) \) is on the graph of \( f^{-1} \)
• The domain of \( f \) becomes the range of \( f^{-1} \), and vice versa
One-to-One Functions:
A function must be one-to-one (each output corresponds to exactly one input) to have an inverse that is also a function
Definition:
A function \( f \) is one-to-one if \( f(a) = f(b) \) implies \( a = b \)
OR: Different inputs always produce different outputs
Horizontal Line Test:
Rule:
If any horizontal line intersects the graph at MORE THAN ONE POINT, the function is NOT one-to-one and does NOT have an inverse function
✓ HAS Inverse (One-to-One):
• Linear functions: \( f(x) = 2x + 3 \)
• Cubic functions: \( f(x) = x^3 \)
• Exponential functions: \( f(x) = 2^x \)
✗ NO Inverse (Not One-to-One):
• Quadratic functions: \( f(x) = x^2 \) (unless domain is restricted)
• Absolute value: \( f(x) = |x| \)
• Even functions generally fail the test
2. Find Values of Inverse Functions from Tables
Method:
Key Concept:
If \( f(a) = b \), then \( f^{-1}(b) = a \)
Steps:
1. Look for the output value in the original function table
2. The corresponding input is the answer
3. Essentially, swap the x and y values
Example:
Given table for f(x):
| x | f(x) |
|---|---|
| 1 | 5 |
| 2 | 8 |
| 3 | 11 |
Find \( f^{-1}(8) \):
Look in the f(x) column for 8
Find corresponding x value: x = 2
Answer: \( f^{-1}(8) = 2 \)
3. Find Values of Inverse Functions from Graphs
Reading Inverse Values from Graphs:
To find \( f^{-1}(a) \) from the graph of f:
1. Locate y = a on the vertical axis
2. Draw a horizontal line to the curve
3. Drop down vertically to the x-axis
4. That x-value is \( f^{-1}(a) \)
Key Insight:
• If point \( (a, b) \) is on the graph of f
• Then \( f(a) = b \) and \( f^{-1}(b) = a \)
• Simply reverse the coordinates!
Example:
If the graph of f passes through points (2, 7), (3, 10), (5, 16):
Then for the inverse function:
\( f^{-1}(7) = 2 \)
\( f^{-1}(10) = 3 \)
\( f^{-1}(16) = 5 \)
4. Graphs of Inverse Functions
Reflection Property:
Key Theorem:
The graph of \( f^{-1} \) is the reflection of the graph of \( f \) across the line \( y = x \)
\[ \text{Graph of } f^{-1} = \text{Graph of } f \text{ reflected over } y = x \]
Why This Works:
• If \( (a, b) \) is on f, then \( (b, a) \) is on \( f^{-1} \)
• Swapping coordinates \( (a, b) \to (b, a) \) is equivalent to reflecting over \( y = x \)
• The line \( y = x \) acts as a "mirror"
To Graph an Inverse Function:
Method 1: Point Reflection
1. Identify key points on the original function
2. Swap the x and y coordinates of each point
3. Plot the new points and connect them
Method 2: Visual Reflection
1. Draw the line \( y = x \)
2. For each point on f, draw a perpendicular to \( y = x \)
3. Extend the same distance on the other side
Important Properties:
• The graphs of f and \( f^{-1} \) are symmetric with respect to \( y = x \)
• Domain of f = Range of \( f^{-1} \)
• Range of f = Domain of \( f^{-1} \)
5. Find Inverse Functions and Relations
Algebraic Method:
Steps to Find \( f^{-1}(x) \):
Step 1: Replace f(x) with y
\( y = f(x) \)
Step 2: Switch x and y
Interchange the variables
Step 3: Solve for y
Isolate y on one side of the equation
Step 4: Replace y with \( f^{-1}(x) \)
This is your inverse function
Step 5: Verify (Optional but Recommended)
Check that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \)
Example 1: Linear Function
Find the inverse of \( f(x) = 3x - 7 \)
Step 1: \( y = 3x - 7 \)
Step 2: \( x = 3y - 7 \)
Step 3: Solve for y:
\( x + 7 = 3y \)
\( y = \frac{x + 7}{3} \)
Step 4: \( f^{-1}(x) = \frac{x + 7}{3} \)
Example 2: Rational Function
Find the inverse of \( f(x) = \frac{2x + 1}{x - 3} \)
Step 1: \( y = \frac{2x + 1}{x - 3} \)
Step 2: \( x = \frac{2y + 1}{y - 3} \)
Step 3: Solve for y:
\( x(y - 3) = 2y + 1 \)
\( xy - 3x = 2y + 1 \)
\( xy - 2y = 3x + 1 \)
\( y(x - 2) = 3x + 1 \)
\( y = \frac{3x + 1}{x - 2} \)
Step 4: \( f^{-1}(x) = \frac{3x + 1}{x - 2} \)
Example 3: Radical Function
Find the inverse of \( f(x) = \sqrt{x - 2} + 3 \), where \( x \geq 2 \)
Step 1: \( y = \sqrt{x - 2} + 3 \)
Step 2: \( x = \sqrt{y - 2} + 3 \)
Step 3: Solve for y:
\( x - 3 = \sqrt{y - 2} \)
\( (x - 3)^2 = y - 2 \)
\( y = (x - 3)^2 + 2 \)
Step 4: \( f^{-1}(x) = (x - 3)^2 + 2 \), where \( x \geq 3 \)
6. Verification of Inverse Functions
Composition Test:
To verify that two functions are inverses, check BOTH compositions:
\[ f(f^{-1}(x)) = x \quad \text{AND} \quad f^{-1}(f(x)) = x \]
BOTH must equal x for the functions to be true inverses
Example Verification:
Verify: \( f(x) = 3x - 7 \) and \( f^{-1}(x) = \frac{x + 7}{3} \)
Check 1: \( f(f^{-1}(x)) \)
\( f\left(\frac{x + 7}{3}\right) = 3\left(\frac{x + 7}{3}\right) - 7 \)
\( = x + 7 - 7 = x \) ✓
Check 2: \( f^{-1}(f(x)) \)
\( f^{-1}(3x - 7) = \frac{(3x - 7) + 7}{3} \)
\( = \frac{3x}{3} = x \) ✓
Both compositions equal x, so they ARE inverse functions!
7. Quick Reference Summary
Key Concepts:
Inverse Definition: \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \)
Horizontal Line Test: Function is one-to-one if no horizontal line hits graph more than once
Graph Property: \( f^{-1} \) is reflection of f over line \( y = x \)
Finding Inverse Algebraically:
1. Replace f(x) with y
2. Switch x and y
3. Solve for y
4. Replace y with \( f^{-1}(x) \)
Domain & Range: Domain of f = Range of \( f^{-1} \)
📚 Study Tips
✓ Always check if a function is one-to-one before finding its inverse
✓ Use horizontal line test to determine if inverse is a function
✓ Remember: (a, b) on f means (b, a) on f⁻¹
✓ Graph of inverse is always reflection over y = x line
✓ Verify your inverse by checking both compositions equal x