Inverse Functions

lessonbunny.com

Answer Key & Explanations

Inverse Functions · Grade 12 · Worksheet 1

1. A function is represented graphically as a cubic curve with points at (-2, -11), (0, 1), and (1, 4). The function is defined as f(x) = ax³ + bx² + cx + d. Find the algebraic expression for the inverse function f⁻¹(x) and evaluate f⁻¹(4). Answer: 1 Solution: Use the given points to find the coefficients a, b, c, and d. For point (-2, -11): a(-2)³ + b(-2)² + c(-2) + d = -11 → -8a + 4b - 2c + d = -11 For point (0, 1): a(0)³ + b(0)² + c(0) + d = 1 → d = 1 For point (1, 4): a(1)³ + b(1)² + c(1) + d = 4 → a + b + c + d = 4 Substitute d = 1 into the…
   Full step-by-step solution

   Step 1: Use the given points to find the coefficients a, b, c, and d. For point (-2, -11): a(-2)³ + b(-2)² + c(-2) + d = -11 → -8a + 4b - 2c + d = -11 For point (0, 1): a(0)³ + b(0)² + c(0) + d = 1 → d = 1 For point (1, 4): a(1)³ + b(1)² + c(1) + d = 4 → a + b + c + d = 4 Step 2: Substitute d = 1 into the equations: -8a + 4b - 2c + 1 = -11 → -8a + 4b - 2c = -12 Divide by -2: 4a - 2b + c = 6 a + b + c + 1 = 4 → a + b + c = 3 Step 3: Solve the system: 4a - 2b + c = 6 a + b + c = 3 Subtract the second equation from the first: (4a - 2b + c) - (a + b + c) = 6 - 3 3a - 3b = 3 → a - b = 1 → a = b + 1 Step 4: We need one more point to solve completely. Let's assume the function passes through (2, 15) as well: a(2)³ + b(2)² + c(2) + 1 = 15 → 8a + 4b + 2c + 1 = 15 → 8a + 4b + 2c = 14 → 4a + 2b + c = 7 Step 5: Now solve the system: 4a - 2b + c = 6 4a + 2b + c = 7 Subtract the first from the second: (4a + 2b + c) - (4a - 2b + c) = 7 - 6 → 4b = 1 → b = 1/4 Then a = b + 1 = 1/4 + 1 = 5/4 From a + b + c = 3: 5/4 + 1/4 + c = 3 → 6/4 + c = 3 → 3/2 + c = 3 → c = 3 - 3/2 = 3/2 Step 6: The function is f(x) = (5/4)x³ + (1/4)x² + (3/2)x + 1 Step 7: To find f⁻¹(4), we need to find x such that f(x) = 4 From our given points, we know f(1) = 4, so f⁻¹(4) = 1 The answer is 1.

2. f(x) = (2x - 3)/(x + 1), find f⁻¹(x) = ? Answer: f⁻¹(x) = (x + 3)/(2 - x) Solution: f(x) = (2x - 3)/(x + 1) y = (2x - 3)/(x + 1) x = (2y - 3)/(y + 1) Multiply both sides by (y + 1): x(y + 1) = 2y - 3 xy + x = 2y - 3 xy - 2y = -3 - x y(x - 2) = -3 - x y = (-3 - x)/(x - 2) Multiply numerator and denominator by -1: y = (x + 3)/(2 - x) f⁻¹(x) = (x + 3)/(2 - x) f⁻¹(x) = (x + 3)/(2 - x)
   Full step-by-step solution

   Let's find the inverse function step by step. We are given: f(x) = (2x - 3)/(x + 1) --- **Step 1: Replace f(x) with y** y = (2x - 3)/(x + 1) --- **Step 2: Swap x and y** To find the inverse, we swap x and y: x = (2y - 3)/(y + 1) --- **Step 3: Solve for y** Multiply both sides by (y + 1): x(y + 1) = 2y - 3 Expand left side: xy + x = 2y - 3 --- **Step 4: Get all terms with y on one side** xy - 2y = -3 - x Factor y on the left: y(x - 2) = -3 - x --- **Step 5: Solve for y** y = (-3 - x)/(x - 2) --- **Step 6: Simplify** Multiply numerator and denominator by -1: y = (x + 3)/(2 - x) --- **Step 7: Write final inverse function** f⁻¹(x) = (x + 3)/(2 - x) --- **Final answer:** f⁻¹(x) = (x + 3)/(2 - x)

3. f(x) = (4x + 1)/(2x - 3), find f⁻¹(x) = ? Answer: (3x + 1)/(2x - 4) Solution: Replace f(x) with y: y = (4x + 1)/(2x - 3) Swap x and y: x = (4y + 1)/(2y - 3) Multiply both sides by (2y - 3): x(2y - 3) = 4y + 1 Distribute: 2xy - 3x = 4y + 1 Move all terms with y to one side: 2xy - 4y = 3x + 1 Factor out y: y(2x - 4) = 3x + 1 Solve for y: y = (3x + 1)/(2x - 4) Replace y with…
   Full step-by-step solution

   Step 1: Replace f(x) with y: y = (4x + 1)/(2x - 3) Step 2: Swap x and y: x = (4y + 1)/(2y - 3) Step 3: Multiply both sides by (2y - 3): x(2y - 3) = 4y + 1 Step 4: Distribute: 2xy - 3x = 4y + 1 Step 5: Move all terms with y to one side: 2xy - 4y = 3x + 1 Step 6: Factor out y: y(2x - 4) = 3x + 1 Step 7: Solve for y: y = (3x + 1)/(2x - 4) Step 8: Replace y with f⁻¹(x): f⁻¹(x) = (3x + 1)/(2x - 4)

4. An environmental scientist is modeling the decay of a radioactive isotope in a contaminated soil sample. The remaining mass M(t) in grams after t years is given by M(t) = (4t + 7)/(t - 2). To determine how long it will take for the isotope to decay to a specific mass, the scientist needs to find the inverse function. What is the inverse function t(M) that gives the time in years when the remaining mass is M grams? Answer: t(M) = (2M + 7)/(M - 4) Solution: Start with the original function: M = (4t + 7)/(t - 2) Multiply both sides by (t - 2): M(t - 2) = 4t + 7 Distribute M: Mt - 2M = 4t + 7 Get all terms with t on one side: Mt - 4t = 2M + 7 Factor out t: t(M - 4) = 2M + 7 Solve for t: t = (2M + 7)/(M - 4) The inverse function is t(M) = (2M + 7)/(M…
   Full step-by-step solution

   Step 1: Start with the original function: M = (4t + 7)/(t - 2) Step 2: Multiply both sides by (t - 2): M(t - 2) = 4t + 7 Step 3: Distribute M: Mt - 2M = 4t + 7 Step 4: Get all terms with t on one side: Mt - 4t = 2M + 7 Step 5: Factor out t: t(M - 4) = 2M + 7 Step 6: Solve for t: t = (2M + 7)/(M - 4) The inverse function is t(M) = (2M + 7)/(M - 4).

5. A function f(x) = (2x - 3)/(x + 1) is graphed on a coordinate plane. The graph shows a curve with a vertical asymptote at x = -1 and a horizontal asymptote at y = 2. Find the algebraic expression for the inverse function f⁻¹(x). Answer: (x + 3)/(2 - x) Solution: f(x) = (2x - 3)/(x + 1) y = (2x - 3)/(x + 1) x = (2y - 3)/(y + 1) Multiply both sides by (y + 1): x(y + 1) = 2y - 3 xy + x = 2y - 3 xy - 2y = -3 - x y(x - 2) = -3 - x y = (-3 - x)/(x - 2) Multiply numerator and denominator by -1: y = (x + 3)/(2 - x) f⁻¹(x) = (x + 3)/(2 - x) f⁻¹(x) = (x + 3)/(2 - x)
   Full step-by-step solution

   Let's find the inverse function step by step. We start with the function: f(x) = (2x - 3)/(x + 1) --- **Step 1: Replace f(x) with y** y = (2x - 3)/(x + 1) --- **Step 2: Swap x and y** To find the inverse, we swap x and y: x = (2y - 3)/(y + 1) --- **Step 3: Solve for y** Multiply both sides by (y + 1): x(y + 1) = 2y - 3 Expand the left side: xy + x = 2y - 3 --- **Step 4: Get all terms with y on one side** xy - 2y = -3 - x Factor y on the left: y(x - 2) = -3 - x --- **Step 5: Solve for y** y = (-3 - x)/(x - 2) --- **Step 6: Simplify the expression** Multiply numerator and denominator by -1: y = (x + 3)/(2 - x) --- **Step 7: Write the inverse function** f⁻¹(x) = (x + 3)/(2 - x) --- **Final answer:** f⁻¹(x) = (x + 3)/(2 - x)

6. An environmental scientist is modeling the decay of a radioactive isotope used in carbon dating. The remaining mass M(t) in grams after t years is given by M(t) = 50 * 2^(-t/5730). To determine how long it takes for the isotope to decay to a specific mass, the scientist needs to find the inverse function. What is the inverse function t(M) that gives the time required for the mass to reach M grams? Answer: t(M) = -5730 * log2(M/50) Solution: Start with the original function: M = 50 * 2^(-t/5730) Swap M and t to find the inverse: t = 50 * 2^(-M/5730) Divide both sides by 50: t/50 = 2^(-M/5730) Take the logarithm base 2 of both sides: log2(t/50) = -M/5730 Multiply both sides by -5730: -5730 * log2(t/50) = M The inverse function is…
   Full step-by-step solution

   Step 1: Start with the original function: M = 50 * 2^(-t/5730) Step 2: Swap M and t to find the inverse: t = 50 * 2^(-M/5730) Step 3: Divide both sides by 50: t/50 = 2^(-M/5730) Step 4: Take the logarithm base 2 of both sides: log2(t/50) = -M/5730 Step 5: Multiply both sides by -5730: -5730 * log2(t/50) = M Step 6: The inverse function is t(M) = -5730 * log2(M/50) The answer is t(M) = -5730 * log2(M/50).

7. A biologist is modeling the population growth of a rare species using the function P(t) = 500e^(0.03t), where t represents time in years and P(t) represents the population size. To determine how long it will take for the population to reach a specific target size, the biologist needs to find the inverse function. Find the inverse function t(P) that gives the time required to reach population P. Answer: t(P) = (ln(P) - ln(500))/0.03 Solution: P(t) = 500 * e^(0.03 * t) We want the inverse function t(P), which tells us the time needed to reach population P.
   Full step-by-step solution

   We start with the population function: P(t) = 500 * e^(0.03 * t) We want the inverse function t(P), which tells us the time needed to reach population P. Step 1: Write the equation with P instead of P(t): P = 500 * e^(0.03 * t) Step 2: Isolate the exponential term by dividing both sides by 500: P / 500 = e^(0.03 * t) Step 3: To solve for t, we take the natural logarithm (ln) of both sides. This is because ln and e are inverse functions. ln(P / 500) = ln(e^(0.03 * t)) Step 4: Use the logarithm property: ln(e^x) = x. So the right-hand side becomes: ln(P / 500) = 0.03 * t Step 5: Solve for t by dividing both sides by 0.03: t = ln(P / 500) / 0.03 Step 6: Use the logarithm property: ln(a / b) = ln(a) - ln(b). So we can write: t = [ln(P) - ln(500)] / 0.03 This is the inverse function t(P). Final answer: t(P) = (ln(P) - ln(500)) / 0.03

8. f(x) = (3x - 2)/(x + 1), find f⁻¹(x) = ? Answer: f⁻¹(x) = (x + 2)/(3 - x) Solution: f(x) = (3x - 2)/(x + 1) y = (3x - 2)/(x + 1) Because for the inverse, x becomes y and y becomes x: x = (3y - 2)/(y + 1) Multiply both sides by (y + 1): x(y + 1) = 3y - 2 xy + x = 3y - 2 xy - 3y = -2 - x y(x - 3) = -2 - x y = (-2 - x)/(x - 3) Multiply numerator and denominator by -1: y = (x +…
   Full step-by-step solution

   Let's find the inverse function step by step. We are given: f(x) = (3x - 2)/(x + 1) --- **Step 1: Replace f(x) with y** y = (3x - 2)/(x + 1) --- **Step 2: Swap x and y** Because for the inverse, x becomes y and y becomes x: x = (3y - 2)/(y + 1) --- **Step 3: Solve for y** Multiply both sides by (y + 1): x(y + 1) = 3y - 2 Distribute x: xy + x = 3y - 2 --- **Step 4: Get all terms with y on one side** xy - 3y = -2 - x Factor y on the left: y(x - 3) = -2 - x --- **Step 5: Solve for y** y = (-2 - x)/(x - 3) --- **Step 6: Simplify** Multiply numerator and denominator by -1: y = (x + 2)/(3 - x) --- **Step 7: Write final answer** f⁻¹(x) = (x + 2)/(3 - x) --- **Final answer:** f⁻¹(x) = (x + 2)/(3 - x)