Inverse Function Graphs

lessonbunny.com

Answer Key & Explanations

Inverse Function Graphs · Grade 12 · Worksheet 3

1. The graph of function f shows f(7) = 2. What is f⁻¹(2)? Answer: 7 Solution: The problem states that f(7) = 2, meaning when the input is 7, the output is 2. 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(7) = 2, meaning when the input is 7, the output is 2. Step 2: For the inverse function f⁻¹, the input and output values are swapped. Step 3: Therefore, if f(7) = 2, then f⁻¹(2) = 7. Step 4: The answer is 7.

2. Aroha analyzes the graph of function f. The graph shows f(7)=11. What is f⁻¹(11)? Answer: 7 Solution: The problem states that f(7)=11, meaning when x=7, the function outputs y=11. For the inverse function f⁻¹, the input and output values are swapped. Since f(7)=11, then f⁻¹(11)=7.
   Full step-by-step solution

   Step 1: The problem states that f(7)=11, meaning when x=7, the function outputs y=11. Step 2: For the inverse function f⁻¹, the input and output values are swapped. Step 3: Since f(7)=11, then f⁻¹(11)=7. Step 4: Therefore, the value of f⁻¹(11) is 7.

3. 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.

4. 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.

5. Mere analyzes the graph of function f and observes f(4)=8. What is f⁻¹(8)? Answer: 4 Solution: The problem states that f(4) = 8, which means when x = 4, y = 8 on the graph of f. For the inverse function f⁻¹, the x and y coordinates are swapped.
   Full step-by-step solution

   Step 1: The problem states that f(4) = 8, which means when x = 4, y = 8 on the graph of f. Step 2: For the inverse function f⁻¹, the x and y coordinates are swapped. Step 3: Therefore, if f(4) = 8, then f⁻¹(8) = 4. Step 4: The answer is 4.

6. A pharmaceutical company is modeling the concentration of a new medication in a patient's bloodstream over time. The concentration C(t) in milligrams per liter is given by the function C(t) = 50e^(-0.2t), where t is time in hours. The therapeutic window for this medication requires maintaining concentrations between 10 mg/L and 30 mg/L. Determine the time interval during which the medication remains within the therapeutic window. Answer: Between approximately 5.58 and 9.16 hours Solution: Exponential decay models are commonly used in pharmacology to describe how drug concentrations decrease over time.
   Full step-by-step solution

   Exponential decay models are commonly used in pharmacology to describe how drug concentrations decrease over time. To find when a function reaches a particular value, you can set the function equal to that value and solve for the variable using logarithms. This approach allows you to determine time intervals where certain conditions are met, which is essential for understanding medication effectiveness and safety windows.

7. A function f(x) is graphed on a coordinate plane as a smooth curve that passes through points (1, 8), (2, 4), (4, 2), and (8, 1). The graph appears to be a hyperbola. The inverse function f⁻¹(x) is the reflection of this curve across the line y = x. What is the value of f⁻¹(4)? Answer: 2 Solution: Step 1: Identify the given points on f(x): (1, 8), (2, 4), (4, 2), and (8, 1) Step 2: Understand that for inverse functions, if (a, b) is on f(x), then (b, a) is on f⁻¹(x) Step 3: Look for a point where f(x) has y-coordinate 4, since we need to find f⁻¹(4) Step 4: From the given points, (2, 4)…
   Full step-by-step solution

   Step 1: Identify the given points on f(x): (1, 8), (2, 4), (4, 2), and (8, 1) Step 2: Understand that for inverse functions, if (a, b) is on f(x), then (b, a) is on f⁻¹(x) Step 3: Look for a point where f(x) has y-coordinate 4, since we need to find f⁻¹(4) Step 4: From the given points, (2, 4) is on f(x), meaning f(2) = 4 Step 5: Therefore, f⁻¹(4) = 2 Step 6: Verify with another point: (4, 2) is on f(x), meaning f(4) = 2, so f⁻¹(2) = 4, confirming the inverse relationship The answer is 2.

8. Aroha analyzes the graph of function f and observes that f(9) = 14. What is f⁻¹(14)? Answer: 9 Solution: The given information is f(9) = 14, which means when x = 9, y = 14 on the graph of f. For inverse functions, the x and y coordinates swap. So if (9, 14) is on f, then (14, 9) is on f⁻¹.
   Full step-by-step solution

   Step 1: The given information is f(9) = 14, which means when x = 9, y = 14 on the graph of f. Step 2: For inverse functions, the x and y coordinates swap. So if (9, 14) is on f, then (14, 9) is on f⁻¹. Step 3: This means f⁻¹(14) = 9. The answer is 9.