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Inverse Function Graphs

Grade 12 · Algebra · Worksheet 2

  1. The graph of function f shows that f(7) = 2. What is f⁻¹(2)? Answer: ______________
  2. Emma analyzes the graph of function f and observes that f(7) = 11. What is f⁻¹(11)? Answer: ______________
  3. Let f(x) = x^3 + 2x^2 - x + 4 be a one-to-one function. If f(1) = 6 and f'(1) = 8, what is the value of (f⁻¹)'(6)? Answer: ______________
  4. The graph of function f shows f(8)=12. What is f⁻¹(12)? Answer: ______________
  5. A function f(x) is graphed on a coordinate plane as a smooth curve passing through points (-3, -8), (-1, 0), (1, 4), and (3, 10). The inverse function f⁻¹(x) is the reflection of this curve across the line y = x. If the point (b, 4) lies on the graph of f(x), what is the value of f⁻¹(4)? Answer: ______________
  6. The graph of function f shows f(4)=8. What is f⁻¹(8)? Answer: ______________
  7. The graph of function f shows f(9)=14 and f(11)=16. Find f⁻¹(14) + f⁻¹(16) = ? Answer: ______________
  8. The graph of function f shows f(9)=14 and f(11)=18. What is f⁻¹(14)? Answer: ______________
  9. The graph of function f shows f(6) = 11. What is f⁻¹(11)? Answer: ______________
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Answer Key & Explanations

Inverse Function Graphs · Grade 12 · Worksheet 2

  1. The graph of function f shows that f(7) = 2. What is f⁻¹(2)? Answer: 7 Solution: The problem states that f(7) = 2, which means when x = 7, f(x) = 2. This corresponds to the point (7,2) on the graph of f. For inverse functions, the coordinates are swapped.
    Full step-by-step solution

    Step 1: The problem states that f(7) = 2, which means when x = 7, f(x) = 2. This corresponds to the point (7,2) on the graph of f. Step 2: For inverse functions, the coordinates are swapped. So if (7,2) is on the graph of f, then (2,7) is on the graph of f⁻¹. Step 3: The point (2,7) on the graph of f⁻¹ means that f⁻¹(2) = 7. Step 4: Therefore, f⁻¹(2) = 7.

  2. Emma analyzes the graph of function f and observes that f(7) = 11. What is f⁻¹(11)? Answer: 7 Solution: The problem states that f(7) = 11, meaning when the input is 7, the output is 11. For the inverse function f⁻¹, the input and output roles are reversed.
    Full step-by-step solution

    Step 1: The problem states that f(7) = 11, meaning when the input is 7, the output is 11. Step 2: For the inverse function f⁻¹, the input and output roles are reversed. Step 3: Therefore, if f(7) = 11, then f⁻¹(11) = 7. Step 4: The answer is 7.

  3. Let f(x) = x^3 + 2x^2 - x + 4 be a one-to-one function. If f(1) = 6 and f'(1) = 8, what is the value of (f⁻¹)'(6)? Answer: 0.125 Solution: Use the formula for the derivative of an inverse function: (f⁻¹)'(b) = 1/f'(a), where f(a) = b We are given f(1) = 6, so a = 1 and b = 6 We are given f'(1) = 8 Apply the formula: (f⁻¹)'(6) = 1/f'(1) = 1/8 1/8 = 0.125 The answer is 0.125.
    Full step-by-step solution

    Step 1: Use the formula for the derivative of an inverse function: (f⁻¹)'(b) = 1/f'(a), where f(a) = b Step 2: We are given f(1) = 6, so a = 1 and b = 6 Step 3: We are given f'(1) = 8 Step 4: Apply the formula: (f⁻¹)'(6) = 1/f'(1) = 1/8 Step 5: 1/8 = 0.125 The answer is 0.125.

  4. The graph of function f shows f(8)=12. What is f⁻¹(12)? Answer: 8 Solution: The given information is f(8) = 12, which means when x = 8, f(x) = 12. For inverse functions, the coordinates are swapped. So if (8,12) is on the graph of f, then (12,8) is on the graph of f⁻¹.
    Full step-by-step solution

    Step 1: The given information is f(8) = 12, which means when x = 8, f(x) = 12. Step 2: For inverse functions, the coordinates are swapped. So if (8,12) is on the graph of f, then (12,8) is on the graph of f⁻¹. Step 3: This means f⁻¹(12) = 8. Step 4: Therefore, the value of f⁻¹(12) is 8.

  5. A function f(x) is graphed on a coordinate plane as a smooth curve passing through points (-3, -8), (-1, 0), (1, 4), and (3, 10). The inverse function f⁻¹(x) is the reflection of this curve across the line y = x. If the point (b, 4) lies on the graph of f(x), what is the value of f⁻¹(4)? Answer: 1 Solution: The problem states that the point (b, 4) lies on the graph of f(x). This means f(b) = 4. We need to find f⁻¹(4).
    Full step-by-step solution

    Step 1: The problem states that the point (b, 4) lies on the graph of f(x). This means f(b) = 4. Step 2: We need to find f⁻¹(4). By definition of inverse functions, if f(b) = 4, then f⁻¹(4) = b. Step 3: Looking at the given points on f(x), we see that the point (1, 4) is on the graph. This means f(1) = 4. Step 4: Therefore, b = 1, and f⁻¹(4) = 1. The answer is 1.

  6. The graph of function f shows f(4)=8. What is f⁻¹(8)? Answer: 4 Solution: The graph shows f(4) = 8, which means the point (4,8) is on the graph of f. For inverse functions, the coordinates are swapped. So if (4,8) is on f, then (8,4) is on f⁻¹.
    Full step-by-step solution

    Step 1: The graph shows f(4) = 8, which means the point (4,8) is on the graph of f. Step 2: For inverse functions, the coordinates are swapped. So if (4,8) is on f, then (8,4) is on f⁻¹. Step 3: This means f⁻¹(8) = 4. The answer is 4.

  7. The graph of function f shows f(9)=14 and f(11)=16. Find f⁻¹(14) + f⁻¹(16) = ? Answer: 20 Solution: From the graph, we know f(9)=14, which means f⁻¹(14)=9 From the graph, we know f(11)=16, which means f⁻¹(16)=11 Add the two inverse function values: f⁻¹(14) + f⁻¹(16) = 9 + 11 9 + 11 = 20 The answer is 20.
    Full step-by-step solution

    Step 1: From the graph, we know f(9)=14, which means f⁻¹(14)=9 Step 2: From the graph, we know f(11)=16, which means f⁻¹(16)=11 Step 3: Add the two inverse function values: f⁻¹(14) + f⁻¹(16) = 9 + 11 Step 4: 9 + 11 = 20 The answer is 20.

  8. The graph of function f shows f(9)=14 and f(11)=18. What is f⁻¹(14)? Answer: 9 Solution: Identify the given information from the graph: f(9)=14 Recall that for inverse functions, if f(a)=b, then f⁻¹(b)=a Apply this to our specific case: since f(9)=14, then f⁻¹(14)=9 The answer is 9
    Full step-by-step solution

    Step 1: Identify the given information from the graph: f(9)=14 Step 2: Recall that for inverse functions, if f(a)=b, then f⁻¹(b)=a Step 3: Apply this to our specific case: since f(9)=14, then f⁻¹(14)=9 Step 4: The answer is 9

  9. The graph of function f shows f(6) = 11. What is f⁻¹(11)? Answer: 6 Solution: The given information is f(6) = 11, which means when x = 6, f(x) = 11. For inverse functions, the coordinates are swapped. So if (6, 11) is on the graph of f, then (11, 6) is on the graph of f⁻¹.
    Full step-by-step solution

    Step 1: The given information is f(6) = 11, which means when x = 6, f(x) = 11. Step 2: For inverse functions, the coordinates are swapped. So if (6, 11) is on the graph of f, then (11, 6) is on the graph of f⁻¹. Step 3: This means f⁻¹(11) = 6. The answer is 6.