Inverse Function Graphs

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Answer Key & Explanations

Inverse Function Graphs · Grade 12 · Worksheet 3

1. The graph of function f shows f(12)=15. What is f⁻¹(15)? Answer: 12 Solution: The problem states that f(12)=15, meaning when x=12, f(x)=15. For the inverse function f⁻¹, the input and output values are swapped.
   Full step-by-step solution

   Step 1: The problem states that f(12)=15, meaning when x=12, f(x)=15. Step 2: For the inverse function f⁻¹, the input and output values are swapped. Step 3: Therefore, if f(12)=15, then f⁻¹(15)=12. Step 4: The answer is 12.

2. Mason is analyzing the graph of a function f(x) in his calculus class. The graph of f passes through the points (2, 7), (7, 12), (12, 17), and (17, 22). If f is one-to-one and invertible, what is the value of f⁻¹(17)? Answer: 12 Solution: The inverse function f⁻¹(y) gives the x-value for which f(x) = y. We are asked for f⁻¹(17), meaning we need to find the x such that f(x) = 17.
   Full step-by-step solution

   Step 1: The inverse function f⁻¹(y) gives the x-value for which f(x) = y. Step 2: We are asked for f⁻¹(17), meaning we need to find the x such that f(x) = 17. Step 3: From the given points on the graph of f, we have (2, 7), (7, 12), (12, 17), and (17, 22). Step 4: The point (12, 17) shows that f(12) = 17. Step 5: Therefore, f⁻¹(17) = 12. The answer is 12.

3. Emma is studying the graph of a function f defined on the interval [-10, 10]. The graph of f is a straight line that passes through the points (-10, -20), (-5, -10), (0, 0), (5, 10), and (10, 20). Using this graph, Emma wants to find the value of f^{-1}(15). What is f^{-1}(15)? Answer: 7.5 Solution: The function f is linear and passes through points like (x, y). The inverse function f^{-1} swaps coordinates: if f(a) = b, then f^{-1}(b) = a. We need f^{-1}(15).
   Full step-by-step solution

   Step 1: Understand the relationship. The function f is linear and passes through points like (x, y). The inverse function f^{-1} swaps coordinates: if f(a) = b, then f^{-1}(b) = a. Step 2: We need f^{-1}(15). This means we need to find the x-value such that f(x) = 15. Step 3: From the graph, the line passes through (5, 10) and (10, 20). Since the line is straight, we can find the equation. The slope is (10 - 0)/(5 - 0) = 10/5 = 2. The y-intercept is 0. So f(x) = 2x. Step 4: Set f(x) = 15: 2x = 15. Step 5: Solve for x: x = 15/2 = 7.5. Step 6: Therefore, f^{-1}(15) = 7.5. The answer is 7.5.

4. 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

5. A pharmaceutical company is modeling the concentration of a new drug in a patient's bloodstream over time using the function C(t) = 80e^(-0.15t), where C is concentration in milligrams per liter and t is time in hours. The therapeutic window for this drug is between 20 mg/L and 60 mg/L. During what time interval is the drug concentration within the therapeutic range? Answer: Between approximately 1.61 hours and 9.24 hours Solution: This requires solving equations involving exponential terms, which typically involves using logarithms to isolate the variable in the exponent.
   Full step-by-step solution

   When modeling exponential decay in real-world scenarios like drug concentration, we often need to find when the function reaches certain threshold values. This requires solving equations involving exponential terms, which typically involves using logarithms to isolate the variable in the exponent. The process involves setting up two separate equations - one for the upper bound and one for the lower bound - then solving each using logarithmic properties.

6. 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.

7. Sophia is analyzing the graph of a one-to-one function f defined on the interval [-10, 10]. On the graph, she identifies the point (6, 1). Later, she needs to find the value of the inverse function at x = 1. Using the relationship between a function and its inverse, what is f⁻¹(1)? Answer: 6 Solution: Recall that if (a, b) is a point on the graph of f, then (b, a) is a point on the graph of f⁻¹. The given point on the graph of f is (6, 1). This means f(6) = 1.
   Full step-by-step solution

   Step 1: Recall that if (a, b) is a point on the graph of f, then (b, a) is a point on the graph of f⁻¹. Step 2: The given point on the graph of f is (6, 1). This means f(6) = 1. Step 3: By the definition of the inverse function, f⁻¹(1) = 6. Step 4: Therefore, the value of f⁻¹(1) is 6. The answer is 6.

8. Isabella analyzes the graph of function f. The graph shows f(7) = 12. What is f⁻¹(12)? Answer: 7 Solution: The graph shows that f(7) = 12, which means when x = 7, y = 12 on the graph of f. For the inverse function f⁻¹, the x and y coordinates swap. So if f(7) = 12, then f⁻¹(12) = 7.
   Full step-by-step solution

   Step 1: The graph shows that f(7) = 12, which means when x = 7, y = 12 on the graph of f. Step 2: For the inverse function f⁻¹, the x and y coordinates swap. So if f(7) = 12, then f⁻¹(12) = 7. Step 3: Therefore, f⁻¹(12) = 7. The answer is 7.