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Function Inverses

Grade 12 · Algebra · Worksheet 1

  1. Aroha is a materials engineer testing the thermal expansion of a new alloy. She models the length of a metal rod (in centimeters) as a function of temperature T (in Celsius) by L(T) = 3T - 7. She suspects that the inverse function, which gives the temperature needed to achieve a certain length, is T(L) = (L + 7)/3. Verify by composition whether L(T) and T(L) are indeed inverse functions for all values in their domains. Answer: ______________
  2. Given f(x) = 5x - 12 and g(x) = (x + 12)/5, verify f(g(23)) = ? Answer: ______________
  3. Isabella is a materials scientist modeling the expansion of a metal rod under heat. The length L (in cm) of the rod at temperature T (in degrees Celsius) is given by L(T) = (3/5)T + 14. Her colleague, Mason, proposes an inverse function T(L) = (5/3)(L - 14) to determine the temperature from a given length. Using function composition, verify whether L(T) and T(L) are indeed inverse functions by computing L(T(L)) and T(L(T)) and checking if both simplify to the original variable. Answer: ______________
  4. Given f(x) = 5x + 10 and g(x) = (x - 10)/5, verify f(g(25)) = ? Answer: ______________
  5. Mason is a structural engineer analyzing the relationship between the load applied to a steel beam and its deflection. The deflection in millimeters is modeled by the function f(x) = 9x - 12, where x is the load in kilonewtons. Mason suspects that the inverse function g(x) = (x + 12)/9 correctly returns the load from a given deflection. Using function composition, verify whether f(x) and g(x) are inverse functions by showing that f(g(x)) = x and g(f(x)) = x. Answer: ______________
  6. Given f(x) = 7x + 12 and g(x) = (x - 12)/7, verify f(g(19)) = ? Answer: ______________
  7. 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: ______________
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Answer Key & Explanations

Function Inverses · Grade 12 · Worksheet 1

  1. Aroha is a materials engineer testing the thermal expansion of a new alloy. She models the length of a metal rod (in centimeters) as a function of temperature T (in Celsius) by L(T) = 3T - 7. She suspects that the inverse function, which gives the temperature needed to achieve a certain length, is T(L) = (L + 7)/3. Verify by composition whether L(T) and T(L) are indeed inverse functions for all values in their domains. Answer: Yes, they are inverse functions because L(T(L)) = L and T(L(T)) = T. Solution: Compute L(T(L)). Start with L(T) = 3T - 7. Substitute T(L) = (L + 7)/3 for T.
    Full step-by-step solution

    Step 1: Compute L(T(L)). Start with L(T) = 3T - 7. Substitute T(L) = (L + 7)/3 for T. So L(T(L)) = 3 * ((L + 7)/3) - 7 = (3(L + 7))/3 - 7 = L + 7 - 7 = L. Step 2: Compute T(L(T)). Start with T(L) = (L + 7)/3. Substitute L(T) = 3T - 7 for L. So T(L(T)) = ((3T - 7) + 7)/3 = (3T)/3 = T. Step 3: Since both compositions give the original input (L and T respectively), the two functions are inverses. Final answer: Yes, they are inverse functions.

  2. Given f(x) = 5x - 12 and g(x) = (x + 12)/5, verify f(g(23)) = ? Answer: 23 Solution: Compute g(23). g(23) = (23 + 12)/5 = 35/5 = 7. Substitute g(23) = 7 into f(x).
    Full step-by-step solution

    Step 1: Compute g(23). g(23) = (23 + 12)/5 = 35/5 = 7. Step 2: Substitute g(23) = 7 into f(x). f(g(23)) = f(7) = 5(7) - 12 = 35 - 12 = 23. Step 3: The result is 23, which equals the original input. This confirms f(g(23)) = 23. The answer is 23.

  3. Isabella is a materials scientist modeling the expansion of a metal rod under heat. The length L (in cm) of the rod at temperature T (in degrees Celsius) is given by L(T) = (3/5)T + 14. Her colleague, Mason, proposes an inverse function T(L) = (5/3)(L - 14) to determine the temperature from a given length. Using function composition, verify whether L(T) and T(L) are indeed inverse functions by computing L(T(L)) and T(L(T)) and checking if both simplify to the original variable. Answer: Yes, they are inverses because L(T(L)) = L and T(L(T)) = T. Solution: Write the given functions. L(T) = (3/5)T + 14. T(L) = (5/3)(L - 14).
    Full step-by-step solution

    Step 1: Write the given functions. L(T) = (3/5)T + 14. T(L) = (5/3)(L - 14). Step 2: Compute L(T(L)). Substitute T(L) into L: L(T(L)) = (3/5) * [ (5/3)(L - 14) ] + 14. Step 3: Multiply: (3/5) * (5/3) = 1, so L(T(L)) = 1*(L - 14) + 14 = L - 14 + 14 = L. Step 4: Compute T(L(T)). Substitute L(T) into T: T(L(T)) = (5/3) * [ (3/5)T + 14 - 14 ] = (5/3) * [ (3/5)T ]. Step 5: Simplify: (5/3)*(3/5)T = 1*T = T. Step 6: Since both compositions yield the original variable (L and T respectively), the functions are inverses. Final answer: Yes, they are inverses.

  4. Given f(x) = 5x + 10 and g(x) = (x - 10)/5, verify f(g(25)) = ? Answer: 25 Solution: Compute g(25) = (25 - 10)/5 = 15/5 = 3 Now compute f(g(25)) = f(3) = 5(3) + 10 = 15 + 10 = 25 The answer is 25.
    Full step-by-step solution

    Step 1: Compute g(25) = (25 - 10)/5 = 15/5 = 3 Step 2: Now compute f(g(25)) = f(3) = 5(3) + 10 = 15 + 10 = 25 The answer is 25.

  5. Mason is a structural engineer analyzing the relationship between the load applied to a steel beam and its deflection. The deflection in millimeters is modeled by the function f(x) = 9x - 12, where x is the load in kilonewtons. Mason suspects that the inverse function g(x) = (x + 12)/9 correctly returns the load from a given deflection. Using function composition, verify whether f(x) and g(x) are inverse functions by showing that f(g(x)) = x and g(f(x)) = x. Answer: Yes, they are inverse functions. Solution: Compute f(g(x)). f(x) = 9x - 12, and g(x) = (x + 12)/9. Substitute g(x) into f: f(g(x)) = 9 * ((x + 12)/9) - 12 = (9/9)(x + 12) - 12 = (x + 12) - 12 = x.
    Full step-by-step solution

    Step 1: Compute f(g(x)). f(x) = 9x - 12, and g(x) = (x + 12)/9. Substitute g(x) into f: f(g(x)) = 9 * ((x + 12)/9) - 12 = (9/9)(x + 12) - 12 = (x + 12) - 12 = x. Step 2: Compute g(f(x)). f(x) = 9x - 12, and g(x) = (x + 12)/9. Substitute f(x) into g: g(f(x)) = ((9x - 12) + 12)/9 = (9x)/9 = x. Step 3: Since f(g(x)) = x and g(f(x)) = x for all x, the functions f and g are inverse functions. Final answer: Yes, they are inverse functions.

  6. Given f(x) = 7x + 12 and g(x) = (x - 12)/7, verify f(g(19)) = ? Answer: 19 Solution: Compute g(19) = (19 - 12)/7 = 7/7 = 1. Now compute f(g(19)) = f(1) = 7(1) + 12 = 7 + 12 = 19. The result is 19, which equals the original input.
    Full step-by-step solution

    Step 1: Compute g(19) = (19 - 12)/7 = 7/7 = 1. Step 2: Now compute f(g(19)) = f(1) = 7(1) + 12 = 7 + 12 = 19. Step 3: The result is 19, which equals the original input. This verifies that f(g(x)) = x for x = 19. The answer is 19.

  7. 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)