Function Inverses

lessonbunny.com

Answer Key & Explanations

Function Inverses · Grade 12 · Worksheet 2

1. Sophia is an astrophysicist modeling the temperature of a star as a function of its core pressure. She defines the temperature function f(x) = 6x - 11, where x represents pressure in megapascals, and the pressure function g(x) = (x + 11)/6. Sophia needs to verify that these two functions are inverses of each other by checking that f(g(x)) = x and g(f(x)) = x for all real x. Perform the composition tests and determine if f and g are indeed inverse functions. Answer: Yes, f and g are inverse functions. Solution: Compute f(g(x)). f(x) = 6x - 11 g(x) = (x + 11)/6 f(g(x)) = 6 * [(x + 11)/6] - 11 = (x + 11) - 11 = x Compute g(f(x)).
   Full step-by-step solution

   Step 1: Compute f(g(x)). f(x) = 6x - 11 g(x) = (x + 11)/6 f(g(x)) = 6 * [(x + 11)/6] - 11 = (x + 11) - 11 = x Step 2: Compute g(f(x)). g(x) = (x + 11)/6 f(x) = 6x - 11 g(f(x)) = [(6x - 11) + 11]/6 = (6x)/6 = x Step 3: Since both f(g(x)) = x and g(f(x)) = x for all real x, the functions f and g are indeed inverses of each other. Final answer: Yes, f and g 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. 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.

4. Given f(x) = 4x - 9 and g(x) = (x + 9)/4, verify f(g(15)) = ? Answer: 15 Solution: Compute g(15). g(15) = (15 + 9)/4 = 24/4 = 6. Now compute f(g(15)) = f(6).
   Full step-by-step solution

   Step 1: Compute g(15). g(15) = (15 + 9)/4 = 24/4 = 6. Step 2: Now compute f(g(15)) = f(6). f(6) = 4(6) - 9 = 24 - 9 = 15. Step 3: The result is 15, which equals the original input. This confirms f(g(15)) = 15. The answer is 15.

5. Aroha is a climate scientist modeling the relationship between atmospheric carbon dioxide concentration (in parts per million, ppm) and global average temperature anomaly (in degrees Celsius). She defines the function f(x) = 7x - 3, where x represents the CO2 concentration in hundreds of ppm, and f(x) gives the temperature anomaly. Her colleague Tane proposes a different function g(x) = (x + 3)/7 to model the same relationship. To determine if f and g are inverse functions, Aroha decides to test them using function composition. Specifically, she will compute f(g(x)) and g(f(x)). Are f(x) = 7x - 3 and g(x) = (x + 3)/7 inverse functions? Answer: Yes, they are inverse functions because f(g(x)) = x and g(f(x)) = x for all x. Solution: Compute f(g(x)). Substitute g(x) = (x + 3)/7 into f(x) = 7x - 3. f(g(x)) = 7 * ((x + 3)/7) - 3.
   Full step-by-step solution

   Step 1: Compute f(g(x)). Substitute g(x) = (x + 3)/7 into f(x) = 7x - 3. f(g(x)) = 7 * ((x + 3)/7) - 3. Simplify: 7 * ((x + 3)/7) = x + 3. Then x + 3 - 3 = x. So f(g(x)) = x. Step 2: Compute g(f(x)). Substitute f(x) = 7x - 3 into g(x) = (x + 3)/7. g(f(x)) = ((7x - 3) + 3)/7. Simplify numerator: 7x - 3 + 3 = 7x. Then (7x)/7 = x. So g(f(x)) = x. Step 3: Since f(g(x)) = x and g(f(x)) = x for all x, f and g are inverse functions. Final answer: Yes, they are inverse functions because f(g(x)) = x and g(f(x)) = x for all x.

6. A cubic function f(x) = x³ + 2 is graphed on a coordinate plane. Its inverse function f⁻¹(x) is also graphed. If you start at the point (1,3) on f(x), 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: (3,1) Solution: Start at point (1,3) on f(x) Move horizontally to the line y = x. This means we keep the same y-coordinate (3) and find where y = x, so we go to point (3,3) Move vertically from (3,3) to f⁻¹(x).
   Full step-by-step solution

   Step 1: Start at point (1,3) on f(x) Step 2: Move horizontally to the line y = x. This means we keep the same y-coordinate (3) and find where y = x, so we go to point (3,3) Step 3: Move vertically from (3,3) to f⁻¹(x). Since f⁻¹(x) is the inverse of f(x), and we're at x = 3, we need to find f⁻¹(3) Step 4: To find f⁻¹(3), solve f(x) = 3: x³ + 2 = 3 → x³ = 1 → x = 1 Step 5: Therefore, f⁻¹(3) = 1, so the point on f⁻¹(x) is (3,1) The final coordinates are (3,1).

7. Given f(x) = 3x³ + 2 and g(x) = ∛((x - 2)/3), verify f(g(29)) = ? Answer: 29 Solution: Compute g(29). g(29) = ∛((29 - 2)/3) = ∛(27/3) = ∛9. Now compute f(g(29)) = f(∛9) = 3(∛9)³ + 2.
   Full step-by-step solution

   Step 1: Compute g(29). g(29) = ∛((29 - 2)/3) = ∛(27/3) = ∛9. Step 2: Now compute f(g(29)) = f(∛9) = 3(∛9)³ + 2. Step 3: Since (∛9)³ = 9, we have 3 × 9 + 2 = 27 + 2 = 29. Step 4: Therefore, f(g(29)) = 29. This confirms that the composition yields the original input, verifying the functions are inverses. The answer is 29.