Function Inverses

Grade 12 · algebra · 113 practice problems · read aloud

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Function Inverses 🔁

A function inverse, denoted f⁻¹(x), essentially "reverses" the action of the original function f(x). If f(a) = b, then f⁻¹(b) = a. It's crucial for solving equations where the variable is trapped inside a function and for understanding the symmetrical relationship between two functions.

Step-by-Step Guide

  1. Replace f(x): Write the function as y = f(x).
  2. Swap x and y: This is the key step! Exchange every x with a y and every y with an x.
  3. Solve for y: Algebraically manipulate the equation to isolate y on one side.
  4. Replace with f⁻¹(x): The new expression for y is your inverse function, f⁻¹(x).
  5. Verify: Check that f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.

Worked Examples

Example 1: Linear Function

Find the inverse of f(x) = 2x - 4.

  1. y = 2x - 4
  2. x = 2y - 4 (Swapped x and y)
  3. x + 4 = 2y → y = (x + 4)/2 (Solved for y)
  4. f⁻¹(x) = (x + 4)/2

Example 2: Rational Function

Find the inverse of g(x) = (x + 1)/(x - 2), where x ≠ 2.

  1. y = (x + 1)/(x - 2)
  2. x = (y + 1)/(y - 2) (Swapped x and y)
  3. x(y - 2) = y + 1 → xy - 2x = y + 1 → xy - y = 2x + 1 → y(x - 1) = 2x + 1
  4. y = (2x + 1)/(x - 1)
  5. g⁻¹(x) = (2x + 1)/(x - 1), x ≠ 1

Common Mistakes to Avoid

❌ Forgetting the Swap: The most common error is solving for x instead of swapping variables first. The inverse is a new function of x.

❌ Ignoring Domain Restrictions: The original function's range becomes the inverse's domain. Always state domain restrictions, especially for rational functions or square roots.

❌ Incorrect Notation: f⁻¹(x) is the inverse function, not 1/f(x) (the reciprocal).

Tips & Tricks

🔍 The Horizontal Line Test: A function has an inverse that is *also a function* if it passes the Horizontal Line Test (no horizontal line intersects the graph more than once).

📈 Graphical Insight: The graph of a function and its inverse are symmetrical about the line y = x. Visualizing this can help you check your work.

🧠 Memory Aid: Think "Inverse = Input/Output Swap."

How to Practice

  • Start with simple linear and quadratic functions, then progress to rational, radical, and exponential/logarithmic pairs.
  • Practice finding the domain and range of both f(x) and f⁻¹(x).
  • Use composition to verify your answers: f(f⁻¹(x)) should always simplify to x.
  • Graph pairs of functions and their inverses on the same axes to see the symmetry about y=x.

Practice problems

6 of the 113, worked through step by step — try them before opening the answer.

1 f(x) = 2x + 3 and g(x) = (x - 3)/2, find f(g(5)) = ?

Hint: To verify if functions are inverses, compose them in both orders and check if the result equals the input value. For different functions, you would substitute the output of one function into the input of the other.

Show the answer

Answer: 5

  1. Find g(5). g(5) = (5 - 3)/2 g(5) = 2/2 g(5) = 1
  2. Now find f(g(5)) = f(1). f(1) = 2*(1) + 3 f(1) = 2 + 3 f(1) = 5
  3. Conclusion. f(g(5)) = 5 Final answer: 5

We are given: f(x) = 2x + 3 g(x) = (x - 3)/2 We want to find f(g(5)).

2 If f(x) = 3x - 7 and g(x) = (x + 7)/3, then f(g(5)) = ?

Hint: To verify if functions are inverses, substitute one function into the other and simplify. For example, if h(x) = 2x + 1 and k(x) = (x - 1)/2, then h(k(3)) should equal 3.

Show the answer

Answer: 5

  1. Compute g(5) g(5) = (5 + 7)/3 g(5) = 12/3 g(5) = 4
  2. Compute f(g(5)) Since g(5) = 4, we have f(g(5)) = f(4) Now use f(x) = 3x - 7: f(4) = 3*4 - 7 f(4) = 12 - 7 f(4) = 5 So the final answer is 5.

We are given: f(x) = 3x - 7 g(x) = (x + 7)/3 We want to compute f(g(5)).

3 f(x) = 3x - 2 and g(x) = (x + 2)/3, verify if f(g(x)) = x

Hint: To verify if functions are inverses, substitute one function into the other and simplify. For example, if h(x) = 2x and k(x) = x/2, then h(k(x)) = 2(x/2) = x.

Show the answer

Answer: f(g(x)) = 3((x + 2)/3) - 2 = x + 2 - 2 = x, yes they are inverses

  1. Write down f(g(x)). This means we take the function g(x) and plug it into f(x) wherever we see x in f(x). So f(g(x)) = 3 * (g(x)) - 2.
  2. Substitute g(x) into that expression. g(x) = (x + 2)/3, so: f(g(x)) = 3 * ((x + 2)/3) - 2.
  3. Simplify the multiplication. 3 * ((x + 2)/3) = x + 2, because the 3 in numerator and denominator cancel. So now we have: f(g(x)) = (x + 2) - 2.
  4. Simplify the subtraction. x + 2 - 2 = x.
  5. Conclusion. We found f(g(x)) = x, which means f and g are inverses of each other. Final answer: f(g(x)) = 3((x + 2)/3) - 2 = x + 2 - 2 = x, yes they are inverses.

We are given two functions: f(x) = 3x - 2 g(x) = (x + 2)/3 We want to verify if f(g(x)) = x.

4 If f(x) = 2x + 5 and g(x) = (x - 5)/2, verify f(g(7)) = ?

Hint: To verify function composition, substitute the output of the inner function into the outer function. For example, if h(x) = x + 1 and k(x) = 2x, then h(k(3)) would be calculated by first finding k(3), then applying h to that result.

Show the answer

Answer: 7

  1. First find g(7) by substituting x = 7 into g(x) = (x - 5)/2 g(7) = (7 - 5)/2 = 2/2 = 1
  2. Now substitute this result into f(x) = 2x + 5 f(g(7)) = f(1) = 2(1) + 5 = 2 + 5 = 7
  3. The final answer is 7 f(g(7)) = 7

5 If f(x) = 2x³ - 1 and g(x) = ∛((x + 1)/2), verify f(g(7)) = ?

Hint: To verify if functions are inverses, substitute one function into the other and simplify. For example, if h(x) = x² and k(x) = √x, then h(k(4)) = (√4)² = 4.

Show the answer

Answer: 7

  1. Start with g(7) = ∛((7 + 1)/2) = ∛(8/2) = ∛4
  2. Now compute f(g(7)) = f(∛4) = 2(∛4)³ - 1
  3. (∛4)³ = 4, so f(g(7)) = 2(4) - 1 = 8 - 1 = 7

The answer is 7.

6 Given f(x) = 5x - 9 and g(x) = (x + 9)/5, verify f(g(14)) = ?

Hint: To verify inverse functions using composition, substitute the inner function into the outer function and simplify step by step. Look for operations that undo each other.

Show the answer

Answer: 14

  1. Compute g(14) = (14 + 9)/5 = 23/5 = 4.6
  2. Now compute f(g(14)) = f(23/5) = 5*(23/5) - 9
  3. Simplify: 5*(23/5) = 23, so 23 - 9 = 14
  4. Therefore, f(g(14)) = 14

The answer is 14.

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