Inverse Functions

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Answer Key & Explanations

Inverse Functions · Grade 12 · Worksheet 1

1. Given f(x) = (1/6)x³ + 1, sketch the graph of f and its inverse f⁻¹ on the same coordinate plane. Verify that the point (6, 37) on f reflects to (37, 6) on f⁻¹ across the line y = x. Answer: The point (37, 6) lies on f⁻¹, confirming the reflection over y = x. Solution: This is a cubic function shifted up by 1 unit and vertically compressed by a factor of 1/6. Yes, (6, 37) is on f.
   Full step-by-step solution

   Step 1: Understand the function f(x) = (1/6)x³ + 1. This is a cubic function shifted up by 1 unit and vertically compressed by a factor of 1/6. Step 2: Verify the given point (6, 37) lies on f(x): f(6) = (1/6)(6³) + 1 = (1/6)(216) + 1 = 36 + 1 = 37. Yes, (6, 37) is on f. Step 3: The inverse function f⁻¹(x) is found by swapping x and y in the equation y = (1/6)x³ + 1 and solving for y: x = (1/6)y³ + 1 x - 1 = (1/6)y³ 6(x - 1) = y³ y = ∛(6(x - 1)) So f⁻¹(x) = ∛(6(x - 1)). Step 4: To verify the reflection, check that (37, 6) lies on f⁻¹: f⁻¹(37) = ∛(6(37 - 1)) = ∛(6 × 36) = ∛216 = 6. Yes, (37, 6) is on f⁻¹. Step 5: The reflection property holds: the point (a, b) on f becomes (b, a) on f⁻¹. Here, (6, 37) reflects to (37, 6) across the line y = x. Step 6: To sketch the graphs: - Plot the line y = x (dashed line, 45° through origin). - Plot f(x) = (1/6)x³ + 1: it passes through (0, 1), (2, 1 + 8/6 ≈ 2.33), (6, 37). It is increasing and symmetric about the origin in shape, shifted up. - Plot f⁻¹(x) = ∛(6(x - 1)): it passes through (1, 0), (2.33, 2), (37, 6). It is the mirror image of f across y = x. - Mark the points (6, 37) and (37, 6) to show the reflection. The answer is that the point (37, 6) lies on f⁻¹, confirming the reflection over y = x.

2. If f(x) = 7x³ + 2 and g(x) is its inverse, then g(247) = ? Answer: ∛35 Solution: Since g(x) is the inverse of f(x), g(247) means find x such that f(x) = 247.
   Full step-by-step solution

   Step 1: Since g(x) is the inverse of f(x), g(247) means find x such that f(x) = 247. Step 2: Set up the equation: 7x³ + 2 = 247 Step 3: Subtract 2 from both sides: 7x³ = 245 Step 4: Divide both sides by 7: x³ = 35 Step 5: Take the cube root of both sides: x = ∛35 Step 6: Therefore, g(247) = ∛35 The answer is ∛35.

3. Dr. Rodriguez is studying the cooling rate of a chemical compound in her laboratory. The temperature T(t) in degrees Celsius after t minutes is modeled by the function T(t) = 80e^(-0.05t) + 20. She needs to determine how long it will take for the compound to cool to 40°C. Find the inverse function T⁻¹(x) that would allow her to calculate the time required to reach any given temperature x. Answer: T⁻¹(x) = -20 * ln((x - 20)/80) Solution: Start with the original function: T(t) = 80e^(-0.05t) + 20 Replace T(t) with x to find the inverse: x = 80e^(-0.05t) + 20 Subtract 20 from both sides: x - 20 = 80e^(-0.05t) Divide both sides by 80: (x - 20)/80 = e^(-0.05t) Take the natural logarithm of both sides: ln((x - 20)/80) = -0.05t…
   Full step-by-step solution

   Step 1: Start with the original function: T(t) = 80e^(-0.05t) + 20 Step 2: Replace T(t) with x to find the inverse: x = 80e^(-0.05t) + 20 Step 3: Subtract 20 from both sides: x - 20 = 80e^(-0.05t) Step 4: Divide both sides by 80: (x - 20)/80 = e^(-0.05t) Step 5: Take the natural logarithm of both sides: ln((x - 20)/80) = -0.05t Step 6: Multiply both sides by -1: -ln((x - 20)/80) = 0.05t Step 7: Divide both sides by 0.05: t = -ln((x - 20)/80)/0.05 Step 8: Simplify the division: t = -20 * ln((x - 20)/80) Step 9: Write the inverse function: T⁻¹(x) = -20 * ln((x - 20)/80) The inverse function is T⁻¹(x) = -20 * ln((x - 20)/80).

4. If f(x) = 3x³ - 5 and g(x) is its inverse, then g(19) = ? Answer: 2 Solution: Since g(x) is the inverse of f(x), we know that g(19) means finding x such that f(x) = 19.
   Full step-by-step solution

   Step 1: Since g(x) is the inverse of f(x), we know that g(19) means finding x such that f(x) = 19. Step 2: Set up the equation: 3x³ - 5 = 19 Step 3: Add 5 to both sides: 3x³ = 24 Step 4: Divide both sides by 3: x³ = 8 Step 5: Take the cube root of both sides: x = 2 Step 6: Therefore, g(19) = 2 The answer is 2.

5. If f(x) = 8x³ + 2 and g(x) is its inverse, then g(514) = ? Answer: 4 Solution: Since g(x) is the inverse of f(x), we know that g(514) means finding x such that f(x) = 514.
   Full step-by-step solution

   Step 1: Since g(x) is the inverse of f(x), we know that g(514) means finding x such that f(x) = 514. Step 2: Set up the equation: 8x³ + 2 = 514 Step 3: Subtract 2 from both sides: 8x³ = 512 Step 4: Divide both sides by 8: x³ = 64 Step 5: Take the cube root of both sides: x = 4 Step 6: Therefore, g(514) = 4 The answer is 4.

6. Given f(x) = 5x³ + 7 and g(x) is its inverse, then g(167) = ? Answer: ∛32 Solution: Since g is the inverse of f, g(167) is the x such that f(x) = 167.
   Full step-by-step solution

   Step 1: Since g is the inverse of f, g(167) is the x such that f(x) = 167. Step 2: Set up the equation: 5x³ + 7 = 167 Step 3: Subtract 7 from both sides: 5x³ = 160 Step 4: Divide both sides by 5: x³ = 32 Step 5: Take the cube root: x = ∛32 Step 6: Therefore, g(167) = ∛32 The answer is ∛32.

7. Given f(x) = 9x - 14, sketch the graph of f and its inverse f⁻¹ on the same coordinate plane. Then, verify that the point (11, 85) on f reflects to (85, 11) on f⁻¹ across the line y = x. Answer: f⁻¹(x) = (x + 14)/9 Solution: Find the inverse function algebraically. Start with f(x) = 9x - 14. Replace f(x) with y: y = 9x - 14.
   Full step-by-step solution

   Step 1: Find the inverse function algebraically. Start with f(x) = 9x - 14. Replace f(x) with y: y = 9x - 14. Swap x and y: x = 9y - 14. Solve for y: x + 14 = 9y, so y = (x + 14)/9. Therefore, f⁻¹(x) = (x + 14)/9. Step 2: Sketch the graph. f(x) = 9x - 14 has slope 9 and y-intercept -14. f⁻¹(x) = (x + 14)/9 has slope 1/9 and y-intercept 14/9 ≈ 1.56. Both lines are reflections of each other across y = x. Step 3: Verify the reflection property for the point (11, 85). Check that (11, 85) lies on f: f(11) = 9(11) - 14 = 99 - 14 = 85. So (11, 85) is on f. Step 4: Check that (85, 11) lies on f⁻¹: f⁻¹(85) = (85 + 14)/9 = 99/9 = 11. So (85, 11) is on f⁻¹. Step 5: The point (11, 85) on f reflects to (85, 11) on f⁻¹ across y = x, confirming the graphical relationship. The answer is f⁻¹(x) = (x + 14)/9.

8. Emma is studying the relationship between two functions in her calculus class. She has function f(x) = (x - 2)^3 + 1 and its inverse f⁻¹(x). When Emma graphs both functions on the same coordinate plane, she notices they intersect at a specific point. Determine the coordinates of this intersection point. Answer: (3, 2) Solution: For any function and its inverse, intersection points occur where f(x) = x, because the inverse function essentially 'undoes' the original function.
   Full step-by-step solution

   For any function and its inverse, intersection points occur where f(x) = x, because the inverse function essentially 'undoes' the original function. This means at intersection points, applying the function to the input gives back the same value as the input when considering the relationship between coordinates. You can verify this by checking if f(a) = a for potential intersection points.

9. Given f(x) = 7x³ + 11 and g(x) is its inverse, then g(60) = ? Answer: ∛7 Solution: Since g(x) is the inverse of f(x), g(60) means find x such that f(x) = 60.
   Full step-by-step solution

   Step 1: Since g(x) is the inverse of f(x), g(60) means find x such that f(x) = 60. Step 2: Set up the equation: 7x³ + 11 = 60 Step 3: Subtract 11 from both sides: 7x³ = 49 Step 4: Divide both sides by 7: x³ = 7 Step 5: Take the cube root of both sides: x = ∛7 Step 6: Therefore, g(60) = ∛7 The answer is ∛7.