Inverse Function Graphs

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Answer Key & Explanations

Inverse Function Graphs · Grade 12 · Worksheet 2

1. The graph of function f shows f(4) = 8. Find f⁻¹(8). Answer: 4 Solution: The problem states that f(4) = 8, meaning when x = 4, f(x) = 8. For inverse functions, the coordinates are swapped. If (4, 8) is a point on f, then (8, 4) is a point on f⁻¹.
   Full step-by-step solution

   Step 1: The problem states that f(4) = 8, meaning when x = 4, f(x) = 8. Step 2: For inverse functions, the coordinates are swapped. If (4, 8) is a point on f, then (8, 4) is a point on f⁻¹. Step 3: Therefore, f⁻¹(8) = 4. The answer is 4.

2. Noah is analyzing the graph of a function f(x) that models the altitude (in hundreds of meters) of a drone over time (in minutes) during a flight. The graph of f passes through the point (12, 20). Noah needs to determine the time at which the drone reaches an altitude of 20 hundred meters, but he realizes that the inverse function f⁻¹ would give him this information directly. What is the value of f⁻¹(20) based on the given point on the graph of f? Answer: 12 Solution: We are given that the point (12, 20) lies on the graph of f. This means f(12) = 20. By definition of an inverse function, if f(a) = b, then f⁻¹(b) = a.
   Full step-by-step solution

   Step 1: We are given that the point (12, 20) lies on the graph of f. This means f(12) = 20. Step 2: By definition of an inverse function, if f(a) = b, then f⁻¹(b) = a. Step 3: Applying this to our values: since f(12) = 20, we have f⁻¹(20) = 12. Step 4: Therefore, the drone reaches an altitude of 20 hundred meters at t = 12 minutes. The answer is 12.

3. A function f(x) is graphed on a coordinate plane. The graph passes through points (-2, 1), (0, 3), and (2, 5), forming a straight line. The inverse function f⁻¹(x) is also graphed. What is the value of f⁻¹(3)? Answer: 0 Solution: We are given that f(x) is a straight line through the points (-2, 1), (0, 3), and (2, 5). We are asked for f⁻¹(3), which means we need the x-value in the original function such that f(x) = 3. Interpret f⁻¹(3).
   Full step-by-step solution

   Step 1: Understand the problem. We are given that f(x) is a straight line through the points (-2, 1), (0, 3), and (2, 5). We are asked for f⁻¹(3), which means we need the x-value in the original function such that f(x) = 3. Step 2: Interpret f⁻¹(3). By definition, f⁻¹(3) is the number a such that f(a) = 3. So we need to find x when f(x) = 3. Step 3: Check the given points. From the points given, we see that when x = 0, f(0) = 3. So f(0) = 3. Step 4: Apply the inverse definition. Since f(0) = 3, then f⁻¹(3) = 0. Step 5: Conclusion. The value of f⁻¹(3) is 0. Final answer: 0

4. Mere analyzes the graph of function f. The graph shows f(12) = 18 and f(15) = 22. What is f⁻¹(18)? Answer: 12 Solution: Identify the given information from the graph: f(12) = 18 and f(15) = 22 For inverse functions, f⁻¹(b) = a when f(a) = b We need to find f⁻¹(18), which means we're looking for the input value that gives an output of 18 in the original function From the graph, f(12) = 18, so when the input is 12,…
   Full step-by-step solution

   Step 1: Identify the given information from the graph: f(12) = 18 and f(15) = 22 Step 2: For inverse functions, f⁻¹(b) = a when f(a) = b Step 3: We need to find f⁻¹(18), which means we're looking for the input value that gives an output of 18 in the original function Step 4: From the graph, f(12) = 18, so when the input is 12, the output is 18 Step 5: Therefore, for the inverse function f⁻¹(18) = 12 Step 6: The answer is 12

5. Matiu analyzes the graph of function f. The graph shows f(12) = 17 and f(15) = 23. Find f⁻¹(17) + f⁻¹(23) = ? Answer: 27 Solution: From the graph, we know f(12) = 17. This means f⁻¹(17) = 12. From the graph, we know f(15) = 23.
   Full step-by-step solution

   Step 1: From the graph, we know f(12) = 17. This means f⁻¹(17) = 12. Step 2: From the graph, we know f(15) = 23. This means f⁻¹(23) = 15. Step 3: Calculate the sum: f⁻¹(17) + f⁻¹(23) = 12 + 15 = 27. The answer is 27.

6. The graph of function f shows f(5)=10. What is f⁻¹(10)? Answer: 5 Solution: The given information is f(5) = 10, which means when x = 5, f(x) = 10. For inverse functions, the coordinates are swapped. So if (5,10) is on the graph of f, then (10,5) is on the graph of f⁻¹.
   Full step-by-step solution

   Step 1: The given information is f(5) = 10, which means when x = 5, f(x) = 10. Step 2: For inverse functions, the coordinates are swapped. So if (5,10) is on the graph of f, then (10,5) is on the graph of f⁻¹. Step 3: This means f⁻¹(10) = 5. The answer is 5.

7. Charlotte is studying the graph of a function f(x) that models the altitude of a drone over time. The graph shows that when the drone has been flying for 9 minutes, its altitude is 27 meters. When the drone has been flying for 14 minutes, its altitude is 12 meters. The function f is one-to-one over the interval shown. What is the value of f⁻¹(27)? Answer: 9 Solution: Understand that f(t) gives altitude at time t. The inverse function f⁻¹(a) gives the time when the altitude is a. From the problem, we know f(9) = 27.
   Full step-by-step solution

   Step 1: Understand that f(t) gives altitude at time t. The inverse function f⁻¹(a) gives the time when the altitude is a. Step 2: From the problem, we know f(9) = 27. This means at time 9 minutes, the altitude is 27 meters. Step 3: By definition of the inverse function, if f(9) = 27, then f⁻¹(27) = 9. Step 4: Therefore, the time when the drone's altitude is 27 meters is 9 minutes. The answer is 9.

8. Sophia is analyzing the graph of a one-to-one function f(x) in her calculus class. The graph passes through the points (1, 6), (6, 11), (11, 16), (16, 21), and (21, 26). Sophia needs to determine the value of f⁻¹(16). What value should she find? Answer: 11 Solution: Understand that f⁻¹(16) asks: 'What x value gives f(x) = 16?' From the given points on the graph of f, identify the point where the y-coordinate is 16. The points are: (1, 6), (6, 11), (11, 16), (16, 21), (21, 26).
   Full step-by-step solution

   Step 1: Understand that f⁻¹(16) asks: 'What x value gives f(x) = 16?' Step 2: From the given points on the graph of f, identify the point where the y-coordinate is 16. Step 3: The points are: (1, 6), (6, 11), (11, 16), (16, 21), (21, 26). Step 4: The point (11, 16) shows that f(11) = 16. Step 5: By definition of inverse functions, if f(11) = 16, then f⁻¹(16) = 11. Step 6: Alternatively, swap the coordinates: (11, 16) becomes (16, 11) for the inverse, so f⁻¹(16) = 11. The answer is 11.