Inverse Functions

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Inverse Functions · Grade 12 · Worksheet 3

1. A marine biologist is studying the cooling rate of ocean water after a volcanic vent becomes inactive. The temperature T(t) in degrees Celsius after t hours is modeled by the function T(t) = 80e^(-0.2t) + 15. The biologist needs to determine how many hours it will take for the water temperature to reach 35°C. Find the inverse function T⁻¹(x) that would allow the biologist to calculate the time required for any given temperature. Answer: T⁻¹(x) = -5 * ln((x - 15)/80) Solution: Start with the original function: T(t) = 80e^(-0.2t) + 15 Replace T(t) with x to find the inverse: x = 80e^(-0.2t) + 15 Subtract 15 from both sides: x - 15 = 80e^(-0.2t) Divide both sides by 80: (x - 15)/80 = e^(-0.2t) Take the natural logarithm of both sides: ln((x - 15)/80) = -0.2t Multiply…
   Full step-by-step solution

   Step 1: Start with the original function: T(t) = 80e^(-0.2t) + 15 Step 2: Replace T(t) with x to find the inverse: x = 80e^(-0.2t) + 15 Step 3: Subtract 15 from both sides: x - 15 = 80e^(-0.2t) Step 4: Divide both sides by 80: (x - 15)/80 = e^(-0.2t) Step 5: Take the natural logarithm of both sides: ln((x - 15)/80) = -0.2t Step 6: Multiply both sides by -1: -ln((x - 15)/80) = 0.2t Step 7: Divide both sides by 0.2: t = -ln((x - 15)/80)/0.2 Step 8: Simplify the division: t = -5 * ln((x - 15)/80) Step 9: Write the inverse function: T⁻¹(x) = -5 * ln((x - 15)/80) The inverse function is T⁻¹(x) = -5 * ln((x - 15)/80).

2. If f(x) = 9x³ + 17 and g(x) is its inverse, then g(764) = ? Answer: ∛83 Solution: Since g(x) is the inverse of f(x), g(764) means find x such that f(x) = 764.
   Full step-by-step solution

   Step 1: Since g(x) is the inverse of f(x), g(764) means find x such that f(x) = 764. Step 2: Set up the equation: 9x³ + 17 = 764 Step 3: Subtract 17 from both sides: 9x³ = 747 Step 4: Divide both sides by 9: x³ = 83 Step 5: Take the cube root of both sides: x = ∛83 Step 6: Therefore, g(764) = ∛83 The answer is ∛83.

3. A marine biologist is studying the cooling rate of ocean water after a volcanic vent becomes inactive. The temperature T in degrees Celsius is modeled by the function T(t) = 80e^(-0.2t) + 15, where t is time in hours. To determine how long it will take for the water to cool to a specific temperature, the biologist needs to find the inverse function. What is the inverse function T⁻¹(x) that gives the time required to reach a temperature of x degrees Celsius? Answer: T⁻¹(x) = -5 * ln((x - 15)/80) Solution: Start with the original function: T(t) = 80e^(-0.2t) + 15 Replace T(t) with x: x = 80e^(-0.2t) + 15 Subtract 15 from both sides: x - 15 = 80e^(-0.2t) Divide both sides by 80: (x - 15)/80 = e^(-0.2t) Take the natural logarithm of both sides: ln((x - 15)/80) = -0.2t Multiply both sides by -1:…
   Full step-by-step solution

   Step 1: Start with the original function: T(t) = 80e^(-0.2t) + 15 Step 2: Replace T(t) with x: x = 80e^(-0.2t) + 15 Step 3: Subtract 15 from both sides: x - 15 = 80e^(-0.2t) Step 4: Divide both sides by 80: (x - 15)/80 = e^(-0.2t) Step 5: Take the natural logarithm of both sides: ln((x - 15)/80) = -0.2t Step 6: Multiply both sides by -1: -ln((x - 15)/80) = 0.2t Step 7: Divide both sides by 0.2: t = -ln((x - 15)/80)/0.2 Step 8: Simplify the division: t = -5 * ln((x - 15)/80) Step 9: Write in inverse function notation: T⁻¹(x) = -5 * ln((x - 15)/80) The inverse function is T⁻¹(x) = -5 * ln((x - 15)/80)

4. If f(x) = 8x³ + 2 and g(x) is its inverse, then g(514) = ? Answer: 4 Solution: Since g(x) is the inverse of f(x), we know that g(514) means finding x such that f(x) = 514.
   Full step-by-step solution

   Step 1: Since g(x) is the inverse of f(x), we know that g(514) means finding x such that f(x) = 514. Step 2: Set up the equation: 8x³ + 2 = 514 Step 3: Subtract 2 from both sides: 8x³ = 512 Step 4: Divide both sides by 8: x³ = 64 Step 5: Take the cube root of both sides: x = 4 Step 6: Therefore, g(514) = 4 The answer is 4.

5. 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^(-0.02t). 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 isotope to decay to any given mass. Answer: M⁻¹(x) = -50 * ln(x/50) Solution: Start with the original function: M(t) = 50e^(-0.02t) Replace M(t) with x to find the inverse: x = 50e^(-0.02t) Divide both sides by 50: x/50 = e^(-0.02t) Take the natural logarithm of both sides: ln(x/50) = ln(e^(-0.02t)) Simplify using logarithm properties: ln(x/50) = -0.02t Solve for t: t =…
   Full step-by-step solution

   Step 1: Start with the original function: M(t) = 50e^(-0.02t) Step 2: Replace M(t) with x to find the inverse: x = 50e^(-0.02t) Step 3: Divide both sides by 50: x/50 = e^(-0.02t) Step 4: Take the natural logarithm of both sides: ln(x/50) = ln(e^(-0.02t)) Step 5: Simplify using logarithm properties: ln(x/50) = -0.02t Step 6: Solve for t: t = ln(x/50) / (-0.02) Step 7: Simplify the expression: t = -50 * ln(x/50) Step 8: Write the inverse function: M⁻¹(x) = -50 * ln(x/50) The inverse function is M⁻¹(x) = -50 * ln(x/50).

6. A function f(x) is graphed on a coordinate plane. The graph passes through points (-2, 1), (0, 3), (2, 5), and (4, 7). If the inverse function f⁻¹(x) exists, what point must lie on its graph? Answer: (3, 0) Solution: We are given points on the graph of f(x): (-2, 1), (0, 3), (2, 5), (4, 7). We are told the inverse function f⁻¹(x) exists. We need to find a point that lies on the graph of f⁻¹(x).
   Full step-by-step solution

   Step 1: Understand the problem. We are given points on the graph of f(x): (-2, 1), (0, 3), (2, 5), (4, 7). We are told the inverse function f⁻¹(x) exists. We need to find a point that lies on the graph of f⁻¹(x). Step 2: Recall the relationship between f(x) and f⁻¹(x). If (a, b) is on the graph of f(x), then (b, a) is on the graph of f⁻¹(x). This is because f(a) = b implies f⁻¹(b) = a. Step 3: Check the given points for f(x). From the points: f(-2) = 1 f(0) = 3 f(2) = 5 f(4) = 7 Step 4: Apply the inverse relationship. For each point (a, b) on f, the point (b, a) is on f⁻¹: From (-2, 1) → (1, -2) is on f⁻¹ From (0, 3) → (3, 0) is on f⁻¹ From (2, 5) → (5, 2) is on f⁻¹ From (4, 7) → (7, 4) is on f⁻¹ Step 5: Identify which of these is the answer. The problem asks: "what point must lie on its graph?" Looking at the options, the correct answer given is (3, 0). From step 4, (3, 0) comes from the point (0, 3) on f(x). Step 6: Verify the function. Notice all points on f(x) satisfy y = x + 3: -2 + 3 = 1 0 + 3 = 3 2 + 3 = 5 4 + 3 = 7 So f(x) = x + 3. Then f⁻¹(x) = x - 3. Check: f⁻¹(3) = 3 - 3 = 0, so (3, 0) is on f⁻¹. Final answer: (3, 0)

7. A marine biologist is studying the cooling rate of ocean water after a volcanic eruption. The temperature T in degrees Celsius as a function of time t in hours is modeled by T(t) = 80e^(-0.2t) + 15. The biologist needs to determine how long it will take for the water temperature to reach 35°C. Find the inverse function t(T) that would allow direct calculation of the time required to reach any given temperature. Answer: t(T) = -5 * ln((T - 15)/80) Solution: Start with the original function: T = 80e^(-0.2t) + 15 Subtract 15 from both sides: T - 15 = 80e^(-0.2t) Divide both sides by 80: (T - 15)/80 = e^(-0.2t) Take the natural logarithm of both sides: ln((T - 15)/80) = -0.2t Multiply both sides by -1: -ln((T - 15)/80) = 0.2t Divide both sides by 0.2:…
   Full step-by-step solution

   Step 1: Start with the original function: T = 80e^(-0.2t) + 15 Step 2: Subtract 15 from both sides: T - 15 = 80e^(-0.2t) Step 3: Divide both sides by 80: (T - 15)/80 = e^(-0.2t) Step 4: Take the natural logarithm of both sides: ln((T - 15)/80) = -0.2t Step 5: Multiply both sides by -1: -ln((T - 15)/80) = 0.2t Step 6: Divide both sides by 0.2: t = -ln((T - 15)/80)/0.2 Step 7: Simplify the division: t = -5 * ln((T - 15)/80) The inverse function is t(T) = -5 * ln((T - 15)/80)