Inverse Functions

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Answer Key & Explanations

Inverse Functions · Grade 12 · Worksheet 2

1. Dr. Rodriguez is studying the decay of a radioactive isotope in a medical application. The remaining mass M(t) in milligrams 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 milligrams. Find the inverse function M⁻¹(x) that would allow her to calculate the time required for the isotope to reach any given mass. Answer: M⁻¹(x) = -ln(x/80)/0.0231 Solution: Start with the original function: M(t) = 80e^(-0.0231t) Replace M(t) with y: y = 80e^(-0.0231t) Switch x and y to find the inverse: 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 Multiply both sides by -1: -ln(x/80) =…
   Full step-by-step solution

   Step 1: Start with the original function: M(t) = 80e^(-0.0231t) Step 2: Replace M(t) with y: y = 80e^(-0.0231t) Step 3: Switch x and y to find the inverse: x = 80e^(-0.0231y) Step 4: Divide both sides by 80: x/80 = e^(-0.0231y) Step 5: Take the natural logarithm of both sides: ln(x/80) = -0.0231y Step 6: Multiply both sides by -1: -ln(x/80) = 0.0231y Step 7: Divide both sides by 0.0231: y = -ln(x/80)/0.0231 Step 8: Write the inverse function: M⁻¹(x) = -ln(x/80)/0.0231 The inverse function is M⁻¹(x) = -ln(x/80)/0.0231

2. The graph of function f(x) is a cubic curve that passes through points (-2, -8), (-1, -1), (0, 0), (1, 1), and (2, 8). The function appears to be strictly increasing and one-to-one. 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(1)? Answer: 1 Solution: Identify the given point on f(x) that has y-coordinate 1. From the problem, f(x) passes through (1, 1), meaning f(1) = 1. For inverse functions, if f(a) = b, then f⁻¹(b) = a.
   Full step-by-step solution

   Step 1: Identify the given point on f(x) that has y-coordinate 1. From the problem, f(x) passes through (1, 1), meaning f(1) = 1. Step 2: For inverse functions, if f(a) = b, then f⁻¹(b) = a. Step 3: Since f(1) = 1, then f⁻¹(1) = 1. Step 4: Verify this makes sense graphically - the point (1, 1) lies on the line y = x, so reflecting it across this line gives the same point. The answer is 1.

3. If f(x) = 4x³ + 2 and g(x) is its inverse, then g(66) = ? Answer: 2 Solution: Since g(x) is the inverse of f(x), we know that g(66) means finding x such that f(x) = 66.
   Full step-by-step solution

   Step 1: Since g(x) is the inverse of f(x), we know that g(66) means finding x such that f(x) = 66. Step 2: Set up the equation: 4x³ + 2 = 66 Step 3: Subtract 2 from both sides: 4x³ = 64 Step 4: Divide both sides by 4: x³ = 16 Step 5: Take the cube root of both sides: x = ∛16 Step 6: Simplify ∛16: ∛(8×2) = ∛8 × ∛2 = 2∛2 Step 7: Therefore, g(66) = 2∛2 The answer is 2∛2.

4. If f(x) = 3x - 5 and g(x) = (x + 5)/3, then g(f(4)) = ? Answer: 4 Solution: Evaluate f(4) where f(x) = 3x - 5 f(4) = 3(4) - 5 = 12 - 5 = 7 Substitute the result into g(x) where g(x) = (x + 5)/3 g(f(4)) = g(7) = (7 + 5)/3 = 12/3 = 4 The answer is 4.
   Full step-by-step solution

   Step 1: Evaluate f(4) where f(x) = 3x - 5 f(4) = 3(4) - 5 = 12 - 5 = 7 Step 2: Substitute the result into g(x) where g(x) = (x + 5)/3 g(f(4)) = g(7) = (7 + 5)/3 = 12/3 = 4 The answer is 4.

5. Dr. Chen is studying the decay of a radioactive isotope in her physics lab. The remaining mass M(t) in grams after t years is modeled by the function M(t) = 50e^(-0.0231t). She needs to determine how many years it will take for the isotope to decay to 25 grams. Find the inverse function M⁻¹(x) that would allow her to calculate the time required for the isotope to reach any given mass. Answer: M⁻¹(x) = -ln(x/50)/0.0231 Solution: Start with the original function: M(t) = 50e^(-0.0231t) Replace M(t) with y: y = 50e^(-0.0231t) Swap x and y to find the inverse: x = 50e^(-0.0231y) Divide both sides by 50: x/50 = e^(-0.0231y) Take the natural logarithm of both sides: ln(x/50) = -0.0231y Solve for y: y = -ln(x/50)/0.0231 Write…
   Full step-by-step solution

   Step 1: Start with the original function: M(t) = 50e^(-0.0231t) Step 2: Replace M(t) with y: y = 50e^(-0.0231t) Step 3: Swap x and y to find the inverse: x = 50e^(-0.0231y) Step 4: Divide both sides by 50: x/50 = e^(-0.0231y) Step 5: Take the natural logarithm of both sides: ln(x/50) = -0.0231y Step 6: Solve for y: y = -ln(x/50)/0.0231 Step 7: Write the inverse function: M⁻¹(x) = -ln(x/50)/0.0231 This inverse function takes a mass value x as input and outputs the time required to reach that mass.

6. Liam is analyzing the relationship between two functions in his calculus class. He has function f(x) = 2x + 3 and its inverse f⁻¹(x). When he 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: Find the inverse function f⁻¹(x). We start with f(x) = 2x + 3. y = 2x + 3 x = 2y + 3 x - 3 = 2y y = (x - 3)/2 So f⁻¹(x) = (x - 3)/2.
   Full step-by-step solution

   Step 1: Find the inverse function f⁻¹(x). We start with f(x) = 2x + 3. To find the inverse, replace f(x) with y: y = 2x + 3 Step 2: Swap x and y: x = 2y + 3 Step 3: Solve for y: x - 3 = 2y y = (x - 3)/2 So f⁻¹(x) = (x - 3)/2. Step 4: Find where f(x) and f⁻¹(x) intersect. Set f(x) = f⁻¹(x): 2x + 3 = (x - 3)/2 Step 5: Multiply both sides by 2 to eliminate the fraction: 2*(2x + 3) = x - 3 4x + 6 = x - 3 Step 6: Solve for x: 4x - x = -3 - 6 3x = -9 x = -3 Step 7: Find the y-coordinate by plugging x = -3 into f(x): f(-3) = 2*(-3) + 3 = -6 + 3 = -3 So the intersection point is (-3, -3). Step 8: Check with f⁻¹(x) to verify: f⁻¹(-3) = (-3 - 3)/2 = (-6)/2 = -3 Yes, both functions give y = -3 when x = -3. Final answer: (-3, -3)

7. A marine biologist is studying the cooling rate of ocean water after a thermal event. The temperature T in degrees Celsius as a function of time t in hours is modeled by T(t) = 80/(t+4) + 10 for t ≥ 0. To determine when the water will reach specific temperatures for coral recovery studies, she needs to find the inverse function. What is the inverse function t(T) that gives the time required to reach a temperature of T degrees Celsius? Answer: t(T) = 80/(T-10) - 4 Solution: To find an inverse function graphically, we reflect the original function across the line y = x. For rational functions, this involves solving the equation for the input variable.
   Full step-by-step solution

   To find an inverse function graphically, we reflect the original function across the line y = x. For rational functions, this involves solving the equation for the input variable. The process demonstrates how inverse functions swap the roles of inputs and outputs, which is visually represented by symmetry across the line y = x on a coordinate plane.