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Function Inverses

Grade 12 · Algebra · Worksheet 1

  1. A function f(x) = 2x + 3 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 graph of f from point (1,5) to the line y = x, then following the reflection to f⁻¹, what is the resulting coordinate point after this composition? Answer: ______________
  2. f(x) = 2x + 3 and g(x) = (x - 3)/2, find f(g(5)) = ? Answer: ______________
  3. A function f(x) = (x - 2)³ + 1 is graphed on a coordinate plane. Its inverse function f⁻¹(x) is also graphed, reflected across the line y = x. If you start at the point (3, 2) on f(x) and follow this path: move horizontally to the line y = x, then vertically to f⁻¹(x), what are the coordinates of the final point reached on f⁻¹(x)? Answer: ______________
  4. Given f(x) = 5x - 9 and g(x) = (x + 9)/5, verify f(g(14)) = ? Answer: ______________
  5. Given f(x) = 6x³ + 1 and g(x) = ∛((x - 1)/6), verify f(g(11)) = ? Answer: ______________
  6. Aroha is an environmental engineer modeling the efficiency of a solar panel system. The energy output E(t) in kilowatt-hours is modeled by the function E(t) = 7t + 9, where t is the number of hours of sunlight. Her colleague, Tane, has derived a function F(t) = (t - 9)/7 that he claims gives the sunlight hours needed to produce t kilowatt-hours. By computing the compositions E(F(t)) and F(E(t)), determine if E(t) and F(t) are inverse functions of each other. Answer: ______________
  7. Noah is designing a digital filter for an audio processing system. The input signal voltage is transformed by the function f(x) = (7x - 3) / 5. To recover the original signal after processing, the system applies an inverse function g(x) = (5x + 3) / 7. Use function composition to verify whether f and g are indeed inverse functions by computing f(g(x)) and g(f(x)) and confirming both equal x. Answer: ______________
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Answer Key & Explanations

Function Inverses · Grade 12 · Worksheet 1

  1. A function f(x) = 2x + 3 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 graph of f from point (1,5) to the line y = x, then following the reflection to f⁻¹, what is the resulting coordinate point after this composition? Answer: (1,1) Solution: Inverse functions have a special geometric relationship where each function is the mirror image of the other across the line y = x.
    Full step-by-step solution

    Inverse functions have a special geometric relationship where each function is the mirror image of the other across the line y = x. When you compose a function with its inverse, the result is the identity function, which means the output equals the original input. This geometric property ensures that following a function and then its inverse brings you back to your starting x-value.

  2. f(x) = 2x + 3 and g(x) = (x - 3)/2, find f(g(5)) = ? Answer: 5 Solution: f(x) = 2x + 3 g(x) = (x - 3)/2 We want to find f(g(5)). Find g(5). g(5) = (5 - 3)/2 g(5) = 2/2 g(5) = 1 Now find f(g(5)) = f(1).
    Full step-by-step solution

    We are given: f(x) = 2x + 3 g(x) = (x - 3)/2 We want to find f(g(5)). Step 1: Find g(5). g(5) = (5 - 3)/2 g(5) = 2/2 g(5) = 1 Step 2: Now find f(g(5)) = f(1). f(1) = 2*(1) + 3 f(1) = 2 + 3 f(1) = 5 Step 3: Conclusion. f(g(5)) = 5 Final answer: 5

  3. A function f(x) = (x - 2)³ + 1 is graphed on a coordinate plane. Its inverse function f⁻¹(x) is also graphed, reflected across the line y = x. If you start at the point (3, 2) on f(x) and follow this path: move horizontally to the line y = x, then vertically to f⁻¹(x), what are the coordinates of the final point reached on f⁻¹(x)? Answer: (2, 3) Solution: Start at point (3, 2) on f(x) Move horizontally to the line y = x.
    Full step-by-step solution

    Step 1: Start at point (3, 2) on f(x) Step 2: Move horizontally to the line y = x. This means we keep the same y-coordinate (2) and find where this horizontal line intersects y = x Step 3: The intersection point is where y = x and y = 2, so the point is (2, 2) Step 4: Now move vertically from (2, 2) to f⁻¹(x). Since f⁻¹(x) is the reflection of f(x) across y = x, moving vertically from the line y = x to f⁻¹(x) means we're essentially reflecting the original point across y = x Step 5: The reflection of (3, 2) across y = x is (2, 3) Step 6: Therefore, the final point reached on f⁻¹(x) is (2, 3) The answer is (2, 3).

  4. Given f(x) = 5x - 9 and g(x) = (x + 9)/5, verify f(g(14)) = ? Answer: 14 Solution: Compute g(14) = (14 + 9)/5 = 23/5 = 4.6 Now compute f(g(14)) = f(23/5) = 5*(23/5) - 9 Simplify: 5*(23/5) = 23, so 23 - 9 = 14 Therefore, f(g(14)) = 14 The answer is 14.
    Full step-by-step solution

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

  5. Given f(x) = 6x³ + 1 and g(x) = ∛((x - 1)/6), verify f(g(11)) = ? Answer: 11 Solution: Compute g(11). g(11) = ∛((11 - 1)/6) = ∛(10/6) = ∛(5/3) Substitute g(11) into f(x). f(g(11)) = f(∛(5/3)) = 6(∛(5/3))³ + 1 Simplify (∛(5/3))³ = 5/3.
    Full step-by-step solution

    Step 1: Compute g(11). g(11) = ∛((11 - 1)/6) = ∛(10/6) = ∛(5/3) Step 2: Substitute g(11) into f(x). f(g(11)) = f(∛(5/3)) = 6(∛(5/3))³ + 1 Step 3: Simplify (∛(5/3))³ = 5/3. So f(g(11)) = 6 × (5/3) + 1 = (6 × 5)/3 + 1 = 30/3 + 1 = 10 + 1 = 11 Step 4: Therefore, f(g(11)) = 11. The answer is 11.

  6. Aroha is an environmental engineer modeling the efficiency of a solar panel system. The energy output E(t) in kilowatt-hours is modeled by the function E(t) = 7t + 9, where t is the number of hours of sunlight. Her colleague, Tane, has derived a function F(t) = (t - 9)/7 that he claims gives the sunlight hours needed to produce t kilowatt-hours. By computing the compositions E(F(t)) and F(E(t)), determine if E(t) and F(t) are inverse functions of each other. Answer: Yes, they are inverse functions Solution: Compute E(F(t)). E(F(t)) = E((t-9)/7) = 7*((t-9)/7) + 9 = (t-9) + 9 = t. Compute F(E(t)).
    Full step-by-step solution

    Step 1: Compute E(F(t)). E(F(t)) = E((t-9)/7) = 7*((t-9)/7) + 9 = (t-9) + 9 = t. Step 2: Compute F(E(t)). F(E(t)) = F(7t+9) = ((7t+9)-9)/7 = (7t)/7 = t. Step 3: Since both compositions equal t, the functions are inverses of each other. Final answer: Yes, they are inverse functions.

  7. Noah is designing a digital filter for an audio processing system. The input signal voltage is transformed by the function f(x) = (7x - 3) / 5. To recover the original signal after processing, the system applies an inverse function g(x) = (5x + 3) / 7. Use function composition to verify whether f and g are indeed inverse functions by computing f(g(x)) and g(f(x)) and confirming both equal x. Answer: Yes, they are inverses. Solution: Compute f(g(x)). Start with g(x) = (5x + 3) / 7. Substitute into f: f(g(x)) = [7 * ((5x + 3)/7) - 3] / 5.
    Full step-by-step solution

    Step 1: Compute f(g(x)). Start with g(x) = (5x + 3) / 7. Substitute into f: f(g(x)) = [7 * ((5x + 3)/7) - 3] / 5. Step 2: Simplify inside the numerator: 7 * ((5x + 3)/7) = 5x + 3. So f(g(x)) = (5x + 3 - 3) / 5 = (5x) / 5 = x. Step 3: Compute g(f(x)). Start with f(x) = (7x - 3) / 5. Substitute into g: g(f(x)) = [5 * ((7x - 3)/5) + 3] / 7. Step 4: Simplify inside the numerator: 5 * ((7x - 3)/5) = 7x - 3. So g(f(x)) = (7x - 3 + 3) / 7 = (7x) / 7 = x. Step 5: Since both compositions equal x, f and g are inverse functions. Final answer: Yes, they are inverses.