Inverse Functions

Finding and understanding inverse functions

Inverse Functions

Definition

f−1f^{-1} is the inverse of ff if: f(f−1(x))=x and f−1(f(x))=xf(f^{-1}(x)) = x \text{ and } f^{-1}(f(x)) = x

The inverse "undoes" what the function does.

Finding an Inverse

Steps:

  1. Write y=f(x)y = f(x)
  2. Swap xx and yy
  3. Solve for yy
  4. Replace yy with f−1(x)f^{-1}(x)

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

  1. y=2x+3y = 2x + 3
  2. x=2y+3x = 2y + 3
  3. x−3=2yx - 3 = 2y, so y=x−32y = \frac{x - 3}{2}
  4. f−1(x)=x−32f^{-1}(x) = \frac{x - 3}{2}

Domain and Range

  • Domain of ff = Range of f−1f^{-1}
  • Range of ff = Domain of f−1f^{-1}

Graphing

The graph of f−1f^{-1} is the reflection of ff over the line y=xy = x.

Horizontal Line Test

ff has an inverse function if and only if no horizontal line intersects the graph more than once.

Verifying

To verify gg is the inverse of ff:

Check that f(g(x))=xf(g(x)) = x AND g(f(x))=xg(f(x)) = x

📚 Practice Problems

1Problem 1easy

❓ Question:

Find the inverse of f(x)=x+7f(x) = x + 7

💡 Show Solution

Step 1: Write as y=x+7y = x + 7

Step 2: Swap xx and yy x=y+7x = y + 7

Step 3: Solve for yy y=x−7y = x - 7

Step 4: Write inverse f−1(x)=x−7f^{-1}(x) = x - 7

Answer: f−1(x)=x−7f^{-1}(x) = x - 7

2Problem 2medium

❓ Question:

Find the inverse of f(x)=x−13f(x) = \frac{x - 1}{3}

💡 Show Solution

Step 1: Write as y=x−13y = \frac{x - 1}{3}

Step 2: Swap xx and yy x=y−13x = \frac{y - 1}{3}

Step 3: Solve for yy 3x=y−13x = y - 1 y=3x+1y = 3x + 1

Step 4: Write inverse f−1(x)=3x+1f^{-1}(x) = 3x + 1

Verify: f(f−1(x))=(3x+1)−13=3x3=xf(f^{-1}(x)) = \frac{(3x + 1) - 1}{3} = \frac{3x}{3} = x ✓

Answer: f−1(x)=3x+1f^{-1}(x) = 3x + 1

3Problem 3hard

❓ Question:

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

💡 Show Solution

Step 1: Write as y=2x+3x−1y = \frac{2x + 3}{x - 1}

Step 2: Swap xx and yy x=2y+3y−1x = \frac{2y + 3}{y - 1}

Step 3: Solve for yy (multiply both sides by denominator) x(y−1)=2y+3x(y - 1) = 2y + 3 xy−x=2y+3xy - x = 2y + 3

Group yy terms: xy−2y=x+3xy - 2y = x + 3 y(x−2)=x+3y(x - 2) = x + 3 y=x+3x−2y = \frac{x + 3}{x - 2}

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

🎴

Practice with Flashcards

Review key concepts with our flashcard system

📖

Browse All Topics

Explore other calculus topics