Inverse Functions

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Answer Key & Explanations

Inverse Functions · Grade 12 · Worksheet 3

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. Given f(x) = 6x³ + 11 and g(x) is its inverse, then g(491) = ? Answer: ∛80 Solution: Since g(x) is the inverse of f(x), g(491) means find x such that f(x) = 491.
   Full step-by-step solution

   Step 1: Since g(x) is the inverse of f(x), g(491) means find x such that f(x) = 491. Step 2: Set up the equation: 6x³ + 11 = 491 Step 3: Subtract 11 from both sides: 6x³ = 480 Step 4: Divide both sides by 6: x³ = 80 Step 5: Take the cube root of both sides: x = ∛80 Step 6: Therefore, g(491) = ∛80 The answer is ∛80.

3. If f(x) = 4x³ + 11 and g(x) is its inverse, then g(75) = ? Answer: 2 Solution: Since g(x) is the inverse of f(x), we know that g(75) means finding x such that f(x) = 75.
   Full step-by-step solution

   Step 1: Since g(x) is the inverse of f(x), we know that g(75) means finding x such that f(x) = 75. Step 2: Set up the equation: 4x³ + 11 = 75 Step 3: Subtract 11 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 the cube root: ∛16 = ∛(8 × 2) = 2∛2 Step 7: Therefore, g(75) = 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. Charlotte is a civil engineer studying the structural load distribution in a suspension bridge. The vertical displacement y (in meters) of a cable at a horizontal distance x (in meters) from the left tower is modeled by the function f(x) = (1/8)(x - 12)^2 + 3 for x in [0, 24]. To quickly determine the horizontal distance corresponding to a given cable height during safety inspections, she needs to find the inverse function restricted to the right half of the cable (where x is from 12 to 24). Find the inverse function f⁻¹(y) for this restricted domain, and describe how the graph of f⁻¹ relates to the graph of f as a reflection. Answer: f⁻¹(y) = 12 + sqrt(8(y - 3)) Solution: Write the function with y: y = (1/8)(x - 12)^2 + 3. Restrict the domain to x >= 12 (right half) so the function is one-to-one. Swap x and y: x = (1/8)(y - 12)^2 + 3.
   Full step-by-step solution

   Step 1: Write the function with y: y = (1/8)(x - 12)^2 + 3. Step 2: Restrict the domain to x >= 12 (right half) so the function is one-to-one. Step 3: Swap x and y: x = (1/8)(y - 12)^2 + 3. Step 4: Solve for y: Subtract 3 from both sides: x - 3 = (1/8)(y - 12)^2. Step 5: Multiply both sides by 8: 8(x - 3) = (y - 12)^2. Step 6: Take the positive square root (since y >= 12): sqrt(8(x - 3)) = y - 12. Step 7: Add 12: y = 12 + sqrt(8(x - 3)). Step 8: Rewrite in inverse notation: f⁻¹(y) = 12 + sqrt(8(y - 3)). Step 9: Graphically, the graph of f⁻¹ is the reflection of the graph of f across the line y = x. For example, if f(12) = 3, then f⁻¹(3) = 12, confirming the symmetry. The final answer is f⁻¹(y) = 12 + sqrt(8(y - 3)).

6. 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.

7. Dr. Rodriguez is studying the decay of a radioactive isotope in a medical sample. The remaining mass M(t) in grams after t days is modeled by the function M(t) = 50e^(-0.0231t). She needs to determine how many days it will take for the sample to decay to 20 grams. Find the inverse function M⁻¹(x) that would allow her to calculate the time required for the sample to reach any specific mass. Answer: ln(x/50)/(-0.0231) Solution: Start with the original function: M(t) = 50e^(-0.0231t) Replace M(t) with x: x = 50e^(-0.0231t) Divide both sides by 50: x/50 = e^(-0.0231t) Take the natural logarithm of both sides: ln(x/50) = ln(e^(-0.0231t)) Simplify using logarithm properties: ln(x/50) = -0.0231t Solve for t: t =…
   Full step-by-step solution

   Step 1: Start with the original function: M(t) = 50e^(-0.0231t) Step 2: Replace M(t) with x: x = 50e^(-0.0231t) Step 3: Divide both sides by 50: x/50 = e^(-0.0231t) Step 4: Take the natural logarithm of both sides: ln(x/50) = ln(e^(-0.0231t)) Step 5: Simplify using logarithm properties: ln(x/50) = -0.0231t Step 6: Solve for t: t = ln(x/50)/(-0.0231) Step 7: Write as the inverse function: M⁻¹(x) = ln(x/50)/(-0.0231) The inverse function is M⁻¹(x) = ln(x/50)/(-0.0231).