Inverse Functions

lessonbunny.com

Answer Key & Explanations

Inverse Functions · Grade 12 · Worksheet 1

1. Dr. Rodriguez is studying the decay of a radioactive isotope in a medical application. The remaining mass M(t) in grams after t days is modeled by the function M(t) = 80e^(-0.0231t). She needs to determine how many days it will take for the isotope to decay to 20 grams. Find the inverse function M⁻¹(x) that would allow her to calculate the time required for the isotope to decay to any given mass. Answer: ln(x/80)/(-0.0231) Solution: Start with the original function: M(t) = 80e^(-0.0231t) To find the inverse, swap the variables: x = 80e^(-0.0231y) Divide both sides by 80: x/80 = e^(-0.0231y) Take the natural logarithm of both sides: ln(x/80) = -0.0231y Divide both sides by -0.0231: y = ln(x/80)/(-0.0231) The inverse function…
   Full step-by-step solution

   Step 1: Start with the original function: M(t) = 80e^(-0.0231t) Step 2: To find the inverse, swap the variables: x = 80e^(-0.0231y) Step 3: Divide both sides by 80: x/80 = e^(-0.0231y) Step 4: Take the natural logarithm of both sides: ln(x/80) = -0.0231y Step 5: Divide both sides by -0.0231: y = ln(x/80)/(-0.0231) Step 6: The inverse function is M⁻¹(x) = ln(x/80)/(-0.0231) The inverse function is ln(x/80)/(-0.0231).

2. Given f(x) = 2x + 5, sketch the graph of f and its inverse f⁻¹ on the same coordinate plane. Then, verify that the point (3, 11) on f reflects to (11, 3) on f⁻¹ across the line y = x. Answer: f⁻¹(x) = (x - 5)/2 Solution: Graph f(x) = 2x + 5. This is a line with slope 2 and y-intercept 5. Plot points: (0, 5) and (1, 7).
   Full step-by-step solution

   Step 1: Graph f(x) = 2x + 5. This is a line with slope 2 and y-intercept 5. Plot points: (0, 5) and (1, 7). Draw the line. Step 2: Draw the line y = x (dashed line) as the mirror. Step 3: Reflect points of f across y = x to get f⁻¹. For (0, 5), the reflection is (5, 0). For (1, 7), the reflection is (7, 1). Plot these points and draw the line through them. Step 4: Find the equation of f⁻¹. Swap x and y in f: x = 2y + 5. Solve for y: 2y = x - 5, so y = (x - 5)/2. Thus f⁻¹(x) = (x - 5)/2. Step 5: Verify the reflection of (3, 11). Check that (3, 11) lies on f: f(3) = 2(3) + 5 = 6 + 5 = 11. So (3, 11) is on f. Its reflection across y = x is (11, 3). Check that (11, 3) lies on f⁻¹: f⁻¹(11) = (11 - 5)/2 = 6/2 = 3. Verified. The answer is f⁻¹(x) = (x - 5)/2.

3. The graph of function f(x) is a smooth curve that passes through points (-2, 16), (-1, 2), (0, 1), (1, 0.5), and (2, 0.25). The curve appears to be exponential in nature. If the inverse function f⁻¹(x) is obtained by reflecting f(x) across the line y = x, what is the value of f⁻¹(2)? Answer: -1 Solution: Step 1: Identify the given points on f(x): (-2, 16), (-1, 2), (0, 1), (1, 0.5), (2, 0.25) Step 2: Understand that for inverse functions, if (a,b) is on f(x), then (b,a) is on f⁻¹(x) Step 3: We need to find f⁻¹(2), which means we're looking for the x-value where f(x) = 2 Step 4: Looking at the…
   Full step-by-step solution

   Step 1: Identify the given points on f(x): (-2, 16), (-1, 2), (0, 1), (1, 0.5), (2, 0.25) Step 2: Understand that for inverse functions, if (a,b) is on f(x), then (b,a) is on f⁻¹(x) Step 3: We need to find f⁻¹(2), which means we're looking for the x-value where f(x) = 2 Step 4: Looking at the points on f(x), we see that when x = -1, f(x) = 2 Step 5: Therefore, f⁻¹(2) = -1 Step 6: Verify: If (-1,2) is on f(x), then (2,-1) must be on f⁻¹(x), so f⁻¹(2) = -1 The answer is -1.

4. If f(x) = 7x³ + 3 and g(x) is its inverse, then g(59) = ? Answer: 2 Solution: Since g(x) is the inverse of f(x), we know that g(59) means finding x such that f(x) = 59.
   Full step-by-step solution

   Step 1: Since g(x) is the inverse of f(x), we know that g(59) means finding x such that f(x) = 59. Step 2: Set up the equation: 7x³ + 3 = 59 Step 3: Subtract 3 from both sides: 7x³ = 56 Step 4: Divide both sides by 7: x³ = 8 Step 5: Take the cube root of both sides: x = 2 Step 6: Therefore, g(59) = 2 The answer is 2.

5. A marine biologist is studying the cooling rate of ocean water after a volcanic eruption. The temperature T in degrees Celsius is modeled by the function T(t) = 20 + 80e^(-0.2t), where t is time in hours. The biologist needs to determine how long it will take for the water temperature to reach 60°C. Find the inverse function T⁻¹(x) that would allow direct calculation of the time required to reach any given temperature. Answer: T⁻¹(x) = -5 * ln((x - 20)/80) Solution: Start with the original function: T(t) = 20 + 80e^(-0.2t) Replace T(t) with x: x = 20 + 80e^(-0.2t) Subtract 20 from both sides: x - 20 = 80e^(-0.2t) Divide both sides by 80: (x - 20)/80 = e^(-0.2t) Take the natural logarithm of both sides: ln((x - 20)/80) = -0.2t Multiply both sides by -1:…
   Full step-by-step solution

   Step 1: Start with the original function: T(t) = 20 + 80e^(-0.2t) Step 2: Replace T(t) with x: x = 20 + 80e^(-0.2t) Step 3: Subtract 20 from both sides: x - 20 = 80e^(-0.2t) Step 4: Divide both sides by 80: (x - 20)/80 = e^(-0.2t) Step 5: Take the natural logarithm of both sides: ln((x - 20)/80) = -0.2t Step 6: Multiply both sides by -1: -ln((x - 20)/80) = 0.2t Step 7: Divide both sides by 0.2: t = -ln((x - 20)/80)/0.2 Step 8: Simplify the division: t = -5 * ln((x - 20)/80) Step 9: Write as the inverse function: T⁻¹(x) = -5 * ln((x - 20)/80) The inverse function is T⁻¹(x) = -5 * ln((x - 20)/80).

6. A function f(x) is graphed on a coordinate plane as a smooth curve that passes through points (-3, -26), (-1, -4), (0, 1), (1, 2), and (2, 7). The function appears to be one-to-one. If the graph of the inverse function f⁻¹(x) is drawn by reflecting f(x) across the line y = x, what point must lie on the graph of f⁻¹(x) given that f(1) = 2? Answer: (2, 1) Solution: The original function f(x) passes through the point (1, 2), meaning f(1) = 2. For inverse functions, if f(a) = b, then f⁻¹(b) = a. Since f(1) = 2, we know f⁻¹(2) = 1.
   Full step-by-step solution

   Step 1: The original function f(x) passes through the point (1, 2), meaning f(1) = 2. Step 2: For inverse functions, if f(a) = b, then f⁻¹(b) = a. Step 3: Since f(1) = 2, we know f⁻¹(2) = 1. Step 4: This means the point (2, 1) must lie on the graph of f⁻¹(x). Step 5: This relationship holds true regardless of the specific function form because it's a fundamental property of inverse functions. The answer is (2, 1).

7. Liam is analyzing the relationship between two functions in his physics experiment. He has function f(x) = 2x + 3 and its inverse f⁻¹(x). When Liam graphs both functions on the same coordinate plane, he notices they intersect at a specific point. Determine the coordinates of this intersection point. Answer: (-3, -3) Solution: f(x) = 2x + 3 To find the inverse, swap x and y and solve for y: Let y = 2x + 3 Swap: x = 2y + 3 x - 3 = 2y y = (x - 3)/2 So f⁻¹(x) = (x - 3)/2 f(x) = f⁻¹(x) 2x + 3 = (x - 3)/2 Multiply both sides by 2: 2(2x + 3) = x - 3 4x + 6 = x - 3 4x - x = -3 - 6 3x = -9 x = -3 Use f(x) = 2x + 3: f(-3) =…
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Find the inverse function f⁻¹(x)** We have f(x) = 2x + 3 To find the inverse, swap x and y and solve for y: Let y = 2x + 3 Swap: x = 2y + 3 Solve for y: x - 3 = 2y y = (x - 3)/2 So f⁻¹(x) = (x - 3)/2 --- **Step 2: Set f(x) equal to f⁻¹(x) to find intersection** At the intersection point, f(x) = f⁻¹(x) 2x + 3 = (x - 3)/2 --- **Step 3: Solve for x** Multiply both sides by 2: 2(2x + 3) = x - 3 4x + 6 = x - 3 4x - x = -3 - 6 3x = -9 x = -3 --- **Step 4: Find y-coordinate** Use f(x) = 2x + 3: f(-3) = 2(-3) + 3 = -6 + 3 = -3 So y = -3. --- **Step 5: Verify with f⁻¹(x)** f⁻¹(-3) = (-3 - 3)/2 = (-6)/2 = -3 Matches. --- **Final Answer:** (-3, -3)