Inverse Function Graphs

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Answer Key & Explanations

Inverse Function Graphs · Grade 12 · Worksheet 1

1. Mere analyzes the graph of function f. The graph shows f(14) = 22. What is f⁻¹(22)? Answer: 14 Solution: The given information is f(14) = 22, which means the point (14, 22) is on the graph of f. For the inverse function f⁻¹, the x-coordinate and y-coordinate are swapped. So if (14, 22) is on f, then (22, 14) is on f⁻¹.
   Full step-by-step solution

   Step 1: The given information is f(14) = 22, which means the point (14, 22) is on the graph of f. Step 2: For the inverse function f⁻¹, the x-coordinate and y-coordinate are swapped. So if (14, 22) is on f, then (22, 14) is on f⁻¹. Step 3: This means f⁻¹(22) = 14. The answer is 14.

2. Kaia is studying the graph of a function f(x) that passes through the points (1, 3), (3, 7), (5, 11), (7, 15), and (9, 19). She needs to determine the value of the inverse function f⁻¹(15). What is f⁻¹(15)? Answer: 7 Solution: Recall that for a function f and its inverse f⁻¹, if f(a) = b, then f⁻¹(b) = a. We are given that the graph of f passes through the point (7, 15). This means f(7) = 15.
   Full step-by-step solution

   Step 1: Recall that for a function f and its inverse f⁻¹, if f(a) = b, then f⁻¹(b) = a. Step 2: We are given that the graph of f passes through the point (7, 15). This means f(7) = 15. Step 3: Therefore, by the definition of the inverse function, f⁻¹(15) = 7. Step 4: Verify by checking the other points: f(1) = 3 so f⁻¹(3) = 1; f(3) = 7 so f⁻¹(7) = 3; f(5) = 11 so f⁻¹(11) = 5; f(9) = 19 so f⁻¹(19) = 9. The answer is 7.

3. Kaia analyzes the graph of function f. The graph shows f(9) = 17 and f(13) = 21. What is f⁻¹(17)? Answer: 9 Solution: The graph shows f(9) = 17, meaning the point (9, 17) is on the graph of f. For the inverse function f⁻¹, the coordinates are swapped. This means f⁻¹(17) = 9.
   Full step-by-step solution

   Step 1: The graph shows f(9) = 17, meaning the point (9, 17) is on the graph of f. Step 2: For the inverse function f⁻¹, the coordinates are swapped. So the point (17, 9) is on the graph of f⁻¹. Step 3: This means f⁻¹(17) = 9. The answer is 9.

4. Hana is analyzing the graph of a function f(x) defined for x between 0 and 8. The graph of f passes through the points (2, 10), (4, 8), (6, 4), and (8, 2). What is the value of f⁻¹(4)? Answer: 6 Solution: Understand that f⁻¹(4) means we need to find the input x such that f(x) = 4. From the given points on the graph of f, we have (6, 4) which means f(6) = 4.
   Full step-by-step solution

   Step 1: Understand that f⁻¹(4) means we need to find the input x such that f(x) = 4. Step 2: From the given points on the graph of f, we have (6, 4) which means f(6) = 4. Step 3: Therefore, f⁻¹(4) = 6. The answer is 6.

5. Kaia is studying the graph of a continuous function f(x) that passes through the points (9, 16) and (12, 25). She knows that f is one-to-one on the interval shown. Using the graph of f, what is the value of f⁻¹(25)? Answer: 12 Solution: The graph of f includes the point (12, 25). This means f(12) = 25. By definition of the inverse function, if f(a) = b, then f⁻¹(b) = a.
   Full step-by-step solution

   Step 1: The graph of f includes the point (12, 25). This means f(12) = 25. Step 2: By definition of the inverse function, if f(a) = b, then f⁻¹(b) = a. Step 3: Here, a = 12 and b = 25, so f⁻¹(25) = 12. Step 4: We can also think of this as swapping the coordinates: the point (12, 25) on f becomes the point (25, 12) on f⁻¹. The answer is 12.

6. Aroha analyzes the graph of function f and observes that f(7)=11. What is f⁻¹(11)? Answer: 7 Solution: The given information states that f(7)=11, meaning when x=7, f(x)=11. For inverse functions, the coordinates are swapped. If (a,b) is on the graph of f, then (b,a) is on the graph of f⁻¹.
   Full step-by-step solution

   Step 1: The given information states that f(7)=11, meaning when x=7, f(x)=11. Step 2: For inverse functions, the coordinates are swapped. If (a,b) is on the graph of f, then (b,a) is on the graph of f⁻¹. Step 3: Since f(7)=11, the point (7,11) is on the graph of f. Step 4: Therefore, the point (11,7) is on the graph of f⁻¹. Step 5: This means f⁻¹(11)=7. The answer is 7.

7. Aroha is analyzing the graph of a one-to-one function f, defined for all real numbers. On the graph, she identifies that the point (3, 11) lies on f. Using the graph, what is the value of f⁻¹(11)? Answer: 3 Solution: The point (3, 11) is on the graph of f, meaning f(3) = 11. For the inverse function f⁻¹, we swap the x and y coordinates. So if (a, b) is on f, then (b, a) is on f⁻¹.
   Full step-by-step solution

   Step 1: The point (3, 11) is on the graph of f, meaning f(3) = 11. Step 2: For the inverse function f⁻¹, we swap the x and y coordinates. So if (a, b) is on f, then (b, a) is on f⁻¹. Step 3: Therefore, f⁻¹(11) = 3. The answer is 3.

8. A pharmaceutical company is modeling the concentration of a new medication in a patient's bloodstream using the function C(t) = 80e^(-0.15t), where t is time in hours and C(t) is concentration in milligrams per liter. The therapeutic window for this medication is between 10 mg/L and 40 mg/L. Determine the time interval during which the medication concentration remains within the therapeutic range. Answer: approximately 6.16 to 18.42 hours Solution: To find when such a quantity reaches a particular value, we use logarithms to solve for the time variable.
   Full step-by-step solution

   Exponential decay models often appear in real-world scenarios like medication concentration, radioactive decay, or cooling processes. To find when such a quantity reaches a particular value, we use logarithms to solve for the time variable. The natural logarithm function is particularly useful when working with base e exponentials, as it allows us to isolate the exponent and solve for time.

9. Sophia analyzes the graph of function f and observes that 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) must be 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) must be on the graph of f⁻¹. Step 3: This means f⁻¹(11) = 6. The answer is 6.