Function Inverses

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Answer Key & Explanations

Function Inverses · Grade 12 · Worksheet 3

1. Kaia is a hydrologist modeling the flow rate of a river after a storm. The flow rate in cubic meters per second is given by the function f(x) = 4x - 9, where x is the water level in meters above flood stage. Her colleague Tane proposes that the inverse function is g(x) = (x + 9)/4. Using function composition, verify whether f(x) and g(x) are inverse functions by computing f(g(x)) and g(f(x)). Answer: Yes, they are inverses because f(g(x)) = x and g(f(x)) = x. Solution: Compute f(g(x)). Substitute g(x) = (x + 9)/4 into f(x) = 4x - 9. f(g(x)) = 4 * ((x + 9)/4) - 9.
   Full step-by-step solution

   Step 1: Compute f(g(x)). Substitute g(x) = (x + 9)/4 into f(x) = 4x - 9. f(g(x)) = 4 * ((x + 9)/4) - 9. Multiply: 4 * ((x + 9)/4) = x + 9. Then subtract 9: (x + 9) - 9 = x. So f(g(x)) = x. Step 2: Compute g(f(x)). Substitute f(x) = 4x - 9 into g(x) = (x + 9)/4. g(f(x)) = ((4x - 9) + 9)/4. Simplify numerator: 4x - 9 + 9 = 4x. Then divide by 4: 4x/4 = x. So g(f(x)) = x. Step 3: Since both f(g(x)) = x and g(f(x)) = x, the functions are inverses of each other. Final answer: Yes, they are inverses because f(g(x)) = x and g(f(x)) = x.

2. Given f(x) = 7x³ + 2 and g(x) = ∛((x - 2)/7), verify f(g(37)) = ? Answer: 37 Solution: Compute g(37) = ∛((37 - 2)/7) = ∛(35/7) = ∛5. Now compute f(g(37)) = f(∛5) = 7(∛5)³ + 2. Since (∛5)³ = 5, we have 7 × 5 + 2 = 35 + 2 = 37.
   Full step-by-step solution

   Step 1: Compute g(37) = ∛((37 - 2)/7) = ∛(35/7) = ∛5. Step 2: Now compute f(g(37)) = f(∛5) = 7(∛5)³ + 2. Step 3: Since (∛5)³ = 5, we have 7 × 5 + 2 = 35 + 2 = 37. Step 4: Therefore, f(g(37)) = 37. The answer is 37.

3. A function f(x) = 2x - 5 is graphed on a coordinate plane. Its inverse function f⁻¹(x) is reflected across the line y = x. If you compose these functions by following the path from point (7,9) on f(x) horizontally to the line y = x, then vertically to f⁻¹(x), what are the coordinates of the final point on f⁻¹(x)? Answer: (9,7) Solution: - \( f(x) = 2x - 5 \) - Inverse \( f^{-1}(x) \) is the reflection of \( f(x) \) across the line \( y = x \). - We start at point \( (7, 9) \) on \( f(x) \). But wait — is \( (7, 9) \) on \( f(x) \)?
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We have: - \( f(x) = 2x - 5 \) - Inverse \( f^{-1}(x) \) is the reflection of \( f(x) \) across the line \( y = x \). - We start at point \( (7, 9) \) on \( f(x) \). But wait — is \( (7, 9) \) on \( f(x) \)? Let's check. --- **Step 2: Check if (7, 9) is on f(x)** \( f(7) = 2(7) - 5 = 14 - 5 = 9 \). Yes, \( f(7) = 9 \), so \( (7, 9) \) is indeed on the graph of \( f(x) \). --- **Step 3: Understand the path described** From \( (7, 9) \) on \( f(x) \): 1. Move horizontally to the line \( y = x \). - Horizontal movement means we keep \( y \) constant at 9, and move to \( y = x \), so we set \( x = 9 \) (since \( y = x \) means \( x = y \)). - So horizontal move: \( (7, 9) \to (9, 9) \). 2. Then move vertically to \( f^{-1}(x) \). - Vertical movement means we keep \( x \) constant at 9, and move to the curve \( f^{-1}(x) \). --- **Step 4: Find \( f^{-1}(x) \)** Let \( y = 2x - 5 \). Swap \( x \) and \( y \): \( x = 2y - 5 \) Solve for \( y \): \( x + 5 = 2y \) \( y = (x + 5)/2 \) So \( f^{-1}(x) = (x + 5)/2 \). --- **Step 5: Move vertically from (9, 9) to \( f^{-1}(x) \)** At \( x = 9 \), \( f^{-1}(9) = (9 + 5)/2 = 14/2 = 7 \). So point on \( f^{-1}(x) \) is \( (9, 7) \). --- **Step 6: Conclusion** The final point on \( f^{-1}(x) \) is \( (9, 7) \). --- **Final answer:** (9, 7)

4. A marine biologist is studying the relationship between water temperature and coral growth rate. The growth rate in millimeters per day is modeled by the function f(x) = 2x + 5, where x represents temperature in degrees Celsius. To analyze the inverse relationship, she needs to find the temperature that corresponds to a specific growth rate. If she composes f with its inverse function f⁻¹, what will be the result of f(f⁻¹(17))? Answer: 17 Solution: We are given the function f(x) = 2x + 5, which gives the coral growth rate for a temperature x. We are told to find f(f⁻¹(17)). f(f⁻¹(a)) = a.
   Full step-by-step solution

   Step 1: Understand the problem We are given the function f(x) = 2x + 5, which gives the coral growth rate for a temperature x. We are told to find f(f⁻¹(17)). Step 2: Recall the property of a function and its inverse By definition, for any value a in the range of f, f(f⁻¹(a)) = a. This is because f⁻¹(a) is the input to f that gives output a, so applying f to that input returns a. Step 3: Apply the property Here, a = 17. So f(f⁻¹(17)) = 17. Step 4: Conclusion No actual calculation of f⁻¹ is needed — the property of inverse functions directly gives the answer. Final answer: 17

5. Given f(x) = 5x - 15 and g(x) = (x + 15)/5, verify f(g(10)) = ? Answer: 10 Solution: Compute g(10) = (10 + 15)/5 = 25/5 = 5. Now compute f(g(10)) = f(5) = 5(5) - 15 = 25 - 15 = 10. The result is 10, which equals the original input.
   Full step-by-step solution

   Step 1: Compute g(10) = (10 + 15)/5 = 25/5 = 5. Step 2: Now compute f(g(10)) = f(5) = 5(5) - 15 = 25 - 15 = 10. Step 3: The result is 10, which equals the original input. This confirms f(g(10)) = 10. The answer is 10.

6. Given f(x) = 5x - 9 and g(x) = (x + 9)/5, verify f(g(11)) = ? Answer: 11 Solution: Evaluate g(11). g(x) = (x + 9)/5, so g(11) = (11 + 9)/5 = 20/5 = 4. Now substitute g(11) = 4 into f(x).
   Full step-by-step solution

   Step 1: Evaluate g(11). g(x) = (x + 9)/5, so g(11) = (11 + 9)/5 = 20/5 = 4. Step 2: Now substitute g(11) = 4 into f(x). f(x) = 5x - 9, so f(4) = 5(4) - 9 = 20 - 9 = 11. Step 3: The result is 11, which equals the original input. This confirms f(g(11)) = 11. The answer is 11.

7. Dr. Chen is studying the cooling rate of a chemical reaction in her lab. The temperature T(t) in degrees Celsius after t minutes is modeled by the function T(t) = 85e^(-0.2t) + 15. She needs to determine the inverse function t(T) to calculate how long it will take for the reaction to cool to 40°C. Verify that T(t) and t(T) are indeed inverse functions by showing their composition equals the identity function. Answer: t(T) = -5ln((T-15)/85) Solution: Inverse functions essentially 'undo' each other's operations. When you compose a function with its inverse, you should get back to your original input value.
   Full step-by-step solution

   Inverse functions essentially 'undo' each other's operations. When you compose a function with its inverse, you should get back to your original input value. For exponential functions, the inverse is typically a logarithmic function, and vice versa. The verification process involves substituting one function into the other and simplifying to see if you get the identity function.