Worksheet 1Worksheet 2Worksheet 3

The download includes this worksheet followed by the answer key with explanations. In the print dialog choose “Save as PDF” to download.

lessonbunny.com
Name: ______________________________ Date: ______________

Function Inverses

Grade 12 · Algebra · Worksheet 2

  1. Matiu is an engineer designing a control system for a robotic arm. The arm's position in centimeters is given by the function f(x) = 4x - 12, where x is the voltage input in volts. The control system uses a calibration function g(x) = (x + 12)/4 to convert the sensor readings back to voltage. Matiu needs to verify that these two functions are inverses of each other by checking their compositions. Compute f(g(x)) and g(f(x)) and determine if they both equal x. Answer: ______________
  2. Liam is designing a new encryption algorithm that relies on a pair of functions to encode and decode messages. The encoding function is f(x) = (3x - 7) / 5, and the decoding function is g(x) = (5x + 7) / 3. To verify his algorithm works correctly, Liam must confirm that f and g are inverse functions. By computing f(g(x)) and g(f(x)), determine if these two functions are indeed inverses of each other. Answer: ______________
  3. f(x) = 2x + 3 and g(x) = (x - 3)/2, find f(g(5)) = ? Answer: ______________
  4. 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: ______________
  5. Aroha is studying the graphs of f(x) = 4x + 9 and g(x) = (x - 9)/4 on a coordinate plane. The graphs are drawn along with the line y = x. Starting from the point (12, 57) on the graph of f(x), Aroha moves horizontally to the line y = x, then vertically to the graph of g(x). What are the coordinates of the final point reached on g(x)? Answer: ______________
  6. Sophia is analyzing two functions that model the growth of a bacterial colony. The functions are f(x) = 6x + 11 and g(x) = (x - 11)/6. She wants to verify whether f and g are inverse functions by checking their compositions. Compute f(g(x)) and g(f(x)), and determine if they both equal x, confirming that f and g are inverses. Answer: ______________
lessonbunny.com

Answer Key & Explanations

Function Inverses · Grade 12 · Worksheet 2

  1. Matiu is an engineer designing a control system for a robotic arm. The arm's position in centimeters is given by the function f(x) = 4x - 12, where x is the voltage input in volts. The control system uses a calibration function g(x) = (x + 12)/4 to convert the sensor readings back to voltage. Matiu needs to verify that these two functions are inverses of each other by checking their compositions. Compute f(g(x)) and g(f(x)) and determine if they both equal x. Answer: Yes, f(g(x)) = x and g(f(x)) = x, so f and g are inverse functions. Solution: Compute f(g(x)). Given f(x) = 4x - 12 and g(x) = (x + 12)/4. Substitute g(x) into f: f(g(x)) = 4 * ((x + 12)/4) - 12.
    Full step-by-step solution

    Step 1: Compute f(g(x)). Given f(x) = 4x - 12 and g(x) = (x + 12)/4. Substitute g(x) into f: f(g(x)) = 4 * ((x + 12)/4) - 12. Simplify: 4 * ((x + 12)/4) = x + 12. Then f(g(x)) = x + 12 - 12 = x. So f(g(x)) = x. Step 2: Compute g(f(x)). Substitute f(x) into g: g(f(x)) = ((4x - 12) + 12)/4. Simplify numerator: 4x - 12 + 12 = 4x. Then g(f(x)) = 4x / 4 = x. So g(f(x)) = x. Step 3: Since both compositions equal x, the functions f(x) = 4x - 12 and g(x) = (x + 12)/4 are inverse functions. The final answer is: Yes, f(g(x)) = x and g(f(x)) = x, so f and g are inverse functions.

  2. Liam is designing a new encryption algorithm that relies on a pair of functions to encode and decode messages. The encoding function is f(x) = (3x - 7) / 5, and the decoding function is g(x) = (5x + 7) / 3. To verify his algorithm works correctly, Liam must confirm that f and g are inverse functions. By computing f(g(x)) and g(f(x)), determine if these two functions are indeed inverses of each other. Answer: Yes, they are inverses Solution: Compute f(g(x)). Substitute g(x) into f(x). f(g(x)) = (3 * ((5x + 7) / 3) - 7) / 5.
    Full step-by-step solution

    Step 1: Compute f(g(x)). Substitute g(x) into f(x). f(g(x)) = (3 * ((5x + 7) / 3) - 7) / 5. Step 2: Simplify inside the numerator. 3 * ((5x + 7) / 3) = 5x + 7. So the numerator becomes (5x + 7) - 7 = 5x. Step 3: Now f(g(x)) = (5x) / 5 = x. Step 4: Compute g(f(x)). Substitute f(x) into g(x). g(f(x)) = (5 * ((3x - 7) / 5) + 7) / 3. Step 5: Simplify inside the numerator. 5 * ((3x - 7) / 5) = 3x - 7. So the numerator becomes (3x - 7) + 7 = 3x. Step 6: Now g(f(x)) = (3x) / 3 = x. Step 7: Since f(g(x)) = x and g(f(x)) = x for all x, the functions are inverse functions. Final answer: Yes, they are inverses.

  3. 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

  4. 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.

  5. Aroha is studying the graphs of f(x) = 4x + 9 and g(x) = (x - 9)/4 on a coordinate plane. The graphs are drawn along with the line y = x. Starting from the point (12, 57) on the graph of f(x), Aroha moves horizontally to the line y = x, then vertically to the graph of g(x). What are the coordinates of the final point reached on g(x)? Answer: (57, 12) Solution: The given point (12, 57) lies on f(x) = 4x + 9. Check: 4(12) + 9 = 48 + 9 = 57. Step 2: Move horizontally from (12, 57) to the line y = x.
    Full step-by-step solution

    Step 1: The given point (12, 57) lies on f(x) = 4x + 9. Check: 4(12) + 9 = 48 + 9 = 57. Step 2: Move horizontally from (12, 57) to the line y = x. On y = x, the y-coordinate stays the same, so y = 57. Since y = x, we have x = 57. The intersection point on y = x is (57, 57). Step 3: From (57, 57), move vertically to g(x). Since g(x) is the inverse of f(x), the reflection of (12, 57) across y = x is (57, 12). Step 4: Verify g(57) = (57 - 9)/4 = 48/4 = 12, so (57, 12) lies on g(x). The final point is (57, 12).

  6. Sophia is analyzing two functions that model the growth of a bacterial colony. The functions are f(x) = 6x + 11 and g(x) = (x - 11)/6. She wants to verify whether f and g are inverse functions by checking their compositions. Compute f(g(x)) and g(f(x)), and determine if they both equal x, confirming that f and g are inverses. Answer: Yes, f and g are inverse functions because f(g(x)) = x and g(f(x)) = x. Solution: Compute f(g(x)). f(x) = 6x + 11, and g(x) = (x - 11)/6. So f(g(x)) = 6 * ((x - 11)/6) + 11.
    Full step-by-step solution

    Step 1: Compute f(g(x)). f(x) = 6x + 11, and g(x) = (x - 11)/6. So f(g(x)) = 6 * ((x - 11)/6) + 11. Simplify: 6 * ((x - 11)/6) = x - 11. Then add 11: (x - 11) + 11 = x. So f(g(x)) = x. Step 2: Compute g(f(x)). g(x) = (x - 11)/6, and f(x) = 6x + 11. So g(f(x)) = ((6x + 11) - 11)/6 = (6x)/6 = 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, f and g are inverse functions because f(g(x)) = x and g(f(x)) = x.