Inverse Function Graphs

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Answer Key & Explanations

Inverse Function Graphs · Grade 12 · Worksheet 1

1. 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.

2. 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.

3. Sophia analyzes the graph of function f and observes that f(8) = 12. What is f⁻¹(12)? Answer: 8 Solution: The problem states that f(8) = 12, which means when x = 8, f(x) = 12. This corresponds to the point (8,12) on the graph of f. For inverse functions, the coordinates are swapped.
   Full step-by-step solution

   Step 1: The problem states that f(8) = 12, which means when x = 8, f(x) = 12. This corresponds to the point (8,12) on the graph of f. Step 2: For inverse functions, the coordinates are swapped. So if (8,12) is on f, then (12,8) is on f⁻¹. Step 3: The notation f⁻¹(12) means we're looking for the y-value when x = 12 on the inverse function, which is 8. Step 4: Therefore, f⁻¹(12) = 8.

4. 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 hours after administration and C(t) is concentration in mg/L. The therapeutic window for this drug is between 15 mg/L and 60 mg/L. Determine the time interval during which the drug concentration remains within the therapeutic range. Answer: approximately 2.31 to 11.59 hours Solution: When working with exponential decay models in pharmacology, we often need to determine time intervals where drug concentrations fall within effective ranges.
   Full step-by-step solution

   When working with exponential decay models in pharmacology, we often need to determine time intervals where drug concentrations fall within effective ranges. This involves solving exponential equations by taking natural logarithms of both sides, which allows us to isolate the time variable. The process requires careful attention to the direction of the inequality when dealing with decreasing functions.

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

   Step 1: The given information is f(6) = 10, which means when x = 6, f(x) = 10. Step 2: For inverse functions, the coordinates are swapped. So if (6, 10) is on the graph of f, then (10, 6) is on the graph of f⁻¹. Step 3: This means f⁻¹(10) = 6. The answer is 6.

6. Emma analyzes the graph of function f. The graph shows f(5)=15 and f(10)=25. What is f⁻¹(15)? Answer: 5 Solution: From the graph, we know f(5)=15 For inverse functions, if f(a)=b, then f⁻¹(b)=a Since f(5)=15, then f⁻¹(15)=5 Therefore, the answer is 5
   Full step-by-step solution

   Step 1: From the graph, we know f(5)=15 Step 2: For inverse functions, if f(a)=b, then f⁻¹(b)=a Step 3: Since f(5)=15, then f⁻¹(15)=5 Step 4: Therefore, the answer is 5

7. A pharmaceutical company is modeling the concentration of a new drug 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 drug is between 15 mg/L and 40 mg/L. Determine the time interval during which the drug concentration remains within the therapeutic range. Answer: Between approximately 1.12 hours and 6.02 hours Solution: C(t) = 50 * e^(-0.2 * t) The therapeutic window is between 15 mg/L and 40 mg/L. We need to find the time interval when 15 < C(t) < 40.
   Full step-by-step solution

   We are given the concentration function: C(t) = 50 * e^(-0.2 * t) The therapeutic window is between 15 mg/L and 40 mg/L. We need to find the time interval when 15 < C(t) < 40. --- **Step 1: Set up the inequalities** Lower bound: 50 * e^(-0.2 * t) > 15 Upper bound: 50 * e^(-0.2 * t) < 40 --- **Step 2: Solve for the lower bound time** 50 * e^(-0.2 * t) = 15 Divide both sides by 50: e^(-0.2 * t) = 15/50 = 3/10 = 0.3 Take natural logarithm of both sides: -0.2 * t = ln(0.3) ln(0.3) ≈ -1.20397 So: -0.2 * t ≈ -1.20397 Divide both sides by -0.2: t ≈ (-1.20397) / (-0.2) t ≈ 6.01985 hours This is the time when concentration drops to 15 mg/L. But since the concentration is decreasing over time, the concentration is above 15 mg/L for t < 6.02 hours. --- **Step 3: Solve for the upper bound time** 50 * e^(-0.2 * t) = 40 Divide both sides by 50: e^(-0.2 * t) = 40/50 = 4/5 = 0.8 Take natural logarithm of both sides: -0.2 * t = ln(0.8) ln(0.8) ≈ -0.223144 So: -0.2 * t ≈ -0.223144 Divide both sides by -0.2: t ≈ (-0.223144) / (-0.2) t ≈ 1.11572 hours This is the time when concentration drops to 40 mg/L from the initial 50 mg/L. The concentration is below 40 mg/L for t > 1.12 hours. --- **Step 4: Interpret the interval** At t = 0, C(0) = 50 mg/L (above therapeutic range). As t increases, C(t) decreases. So C(t) enters the therapeutic range when it drops to 40 mg/L at t ≈ 1.12 hours. C(t) leaves the therapeutic range when it drops below 15 mg/L at t ≈ 6.02 hours. Thus the time interval is: 1.12 hours < t < 6.02 hours. --- **Final Answer:** Between approximately 1.12 hours and 6.02 hours.

8. A function f(x) is graphed on a coordinate plane as a smooth curve passing through points (-3, -8), (-1, -2), (1, 0), (2, 1), and (4, 3). The inverse function f⁻¹(x) is the reflection of this curve across the line y = x. If the point (b, -2) lies on the graph of f⁻¹(x), what is the value of b? Answer: -1 Solution: The point (b, -2) lies on the graph of f⁻¹(x), which means f⁻¹(b) = -2. Since f⁻¹(b) = -2, applying f to both sides gives f(-2) = b. Look at the original function f(x) and find where f(x) = -2.
   Full step-by-step solution

   Step 1: The point (b, -2) lies on the graph of f⁻¹(x), which means f⁻¹(b) = -2. Step 2: Since f⁻¹(b) = -2, applying f to both sides gives f(-2) = b. Step 3: Look at the original function f(x) and find where f(x) = -2. Step 4: From the given points on f(x), we see that f(-1) = -2. Step 5: Therefore, b = -1. The answer is -1.