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Function Inverses

Grade 12 · Algebra · Worksheet 3

  1. 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. 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: ______________
  3. Noah is a financial analyst studying the relationship between two economic models. The first model describes the cost of manufacturing x units of a product as f(x) = (6x - 1) / 5. The second model describes the number of units produced based on a budget of y dollars as g(y) = (5y + 1) / 6. Noah wants to verify that these two functions are inverses of each other, meaning that applying one after the other returns the original input. Use function composition to determine if f(g(x)) = x and g(f(x)) = x hold for all values of x. Answer: ______________
  4. Given f(x) = 3e^(2x) - 1 and g(x) = (1/2)ln((x + 1)/3), determine the value of f(g(5)) + g(f(0)). Answer: ______________
  5. Given f(x) = 7x - 12 and g(x) = (x + 12)/7, verify f(g(27)) = ? Answer: ______________
  6. Mason is an aerospace engineer designing a satellite's thermal control system. The temperature regulation is modeled by two functions: f(x) = 4x - 9 and g(x) = (x + 9)/4, where x represents the electrical current in amperes. To verify that these functions are inverses, Mason must show that f(g(x)) = x and g(f(x)) = x. Perform these compositions and determine whether f and g are inverse functions. Answer: ______________
  7. Given f(x) = 4x³ + 11 and g(x) = ∛((x - 11)/4), verify f(g(27)) = ? Answer: ______________
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Answer Key & Explanations

Function Inverses · Grade 12 · Worksheet 3

  1. 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).

  2. 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.

  3. Noah is a financial analyst studying the relationship between two economic models. The first model describes the cost of manufacturing x units of a product as f(x) = (6x - 1) / 5. The second model describes the number of units produced based on a budget of y dollars as g(y) = (5y + 1) / 6. Noah wants to verify that these two functions are inverses of each other, meaning that applying one after the other returns the original input. Use function composition to determine if f(g(x)) = x and g(f(x)) = x hold for all values of x. Answer: Yes, they are inverses. Solution: Find f(g(x)). Given f(x) = (6x - 1) / 5 and g(x) = (5x + 1) / 6.
    Full step-by-step solution

    Step 1: Find f(g(x)). Given f(x) = (6x - 1) / 5 and g(x) = (5x + 1) / 6. Substitute g(x) into f: f(g(x)) = (6 * g(x) - 1) / 5 f(g(x)) = (6 * ((5x + 1) / 6) - 1) / 5 Simplify inside the numerator: 6 * ((5x + 1) / 6) = 5x + 1 So f(g(x)) = ((5x + 1) - 1) / 5 f(g(x)) = (5x) / 5 = x Step 2: Find g(f(x)). Substitute f(x) into g: g(f(x)) = (5 * f(x) + 1) / 6 g(f(x)) = (5 * ((6x - 1) / 5) + 1) / 6 Simplify inside the numerator: 5 * ((6x - 1) / 5) = 6x - 1 So g(f(x)) = ((6x - 1) + 1) / 6 g(f(x)) = (6x) / 6 = x Step 3: Since f(g(x)) = x and g(f(x)) = x for all x, the two functions are inverses of each other. Final answer: Yes, they are inverses.

  4. Given f(x) = 3e^(2x) - 1 and g(x) = (1/2)ln((x + 1)/3), determine the value of f(g(5)) + g(f(0)). Answer: 5 Solution: Check if f and g are inverses by computing f(g(x)) f(g(x)) = 3e^(2 * (1/2)ln((x + 1)/3)) - 1 = 3e^(ln((x + 1)/3)) - 1 = 3 * ((x + 1)/3) - 1 = (x + 1) - 1 = x Check if f and g are inverses by computing g(f(x)) g(f(x)) = (1/2)ln((3e^(2x) - 1 + 1)/3) = (1/2)ln((3e^(2x))/3) = (1/2)ln(e^(2x)) =…
    Full step-by-step solution

    Step 1: Check if f and g are inverses by computing f(g(x)) f(g(x)) = 3e^(2 * (1/2)ln((x + 1)/3)) - 1 = 3e^(ln((x + 1)/3)) - 1 = 3 * ((x + 1)/3) - 1 = (x + 1) - 1 = x Step 2: Check if f and g are inverses by computing g(f(x)) g(f(x)) = (1/2)ln((3e^(2x) - 1 + 1)/3) = (1/2)ln((3e^(2x))/3) = (1/2)ln(e^(2x)) = (1/2)(2x) = x Step 3: Since f(g(x)) = x and g(f(x)) = x, the functions are inverses Step 4: Evaluate f(g(5)) Since f and g are inverses, f(g(5)) = 5 Step 5: Evaluate g(f(0)) Since f and g are inverses, g(f(0)) = 0 Step 6: Compute f(g(5)) + g(f(0)) = 5 + 0 = 5 The answer is 5.

  5. Given f(x) = 7x - 12 and g(x) = (x + 12)/7, verify f(g(27)) = ? Answer: 27 Solution: Evaluate g(27). g(27) = (27 + 12)/7 = 39/7. Substitute g(27) = 39/7 into f(x).
    Full step-by-step solution

    Step 1: Evaluate g(27). g(27) = (27 + 12)/7 = 39/7. Step 2: Substitute g(27) = 39/7 into f(x). f(g(27)) = f(39/7) = 7*(39/7) - 12. Step 3: Simplify: 7*(39/7) = 39, so 39 - 12 = 27. Step 4: Therefore, f(g(27)) = 27, confirming the composition yields the original input. The answer is 27.

  6. Mason is an aerospace engineer designing a satellite's thermal control system. The temperature regulation is modeled by two functions: f(x) = 4x - 9 and g(x) = (x + 9)/4, where x represents the electrical current in amperes. To verify that these functions are inverses, Mason must show that f(g(x)) = x and g(f(x)) = x. Perform these compositions and determine whether f and g are inverse functions. Answer: Yes, f and g are inverse functions. Solution: Compute f(g(x)). Given f(x) = 4x - 9 and g(x) = (x + 9)/4. Substitute g(x) into f: f(g(x)) = 4 * ((x + 9)/4) - 9.
    Full step-by-step solution

    Step 1: Compute f(g(x)). Given f(x) = 4x - 9 and g(x) = (x + 9)/4. Substitute g(x) into f: f(g(x)) = 4 * ((x + 9)/4) - 9. Simplify: 4 * ((x + 9)/4) = x + 9. Then f(g(x)) = (x + 9) - 9 = x. Step 2: Compute g(f(x)). Substitute f(x) into g: g(f(x)) = ((4x - 9) + 9)/4. Simplify numerator: (4x - 9) + 9 = 4x. Then g(f(x)) = (4x)/4 = x. Since both compositions equal x, f and g are inverse functions. Final answer: Yes, f and g are inverse functions.

  7. Given f(x) = 4x³ + 11 and g(x) = ∛((x - 11)/4), verify f(g(27)) = ? Answer: 27 Solution: Evaluate g(27). g(27) = ∛((27 - 11)/4) = ∛(16/4) = ∛4 Substitute the result into f(x). f(g(27)) = f(∛4) = 4(∛4)³ + 11 Simplify (∛4)³.
    Full step-by-step solution

    Step 1: Evaluate g(27). g(27) = ∛((27 - 11)/4) = ∛(16/4) = ∛4 Step 2: Substitute the result into f(x). f(g(27)) = f(∛4) = 4(∛4)³ + 11 Step 3: Simplify (∛4)³. Since (∛a)³ = a, we have (∛4)³ = 4. Step 4: Complete the calculation. f(g(27)) = 4 × 4 + 11 = 16 + 11 = 27 The answer is 27.