Inverse Functions

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Inverse Functions · Grade 12 · Worksheet 3

1. Hana is a marine biologist modeling the water temperature in a geothermal vent ecosystem. The temperature T(d) in degrees Celsius at depth d meters below the ocean surface is given by the function T(d) = (12d + 15)/(d + 3). To determine the depth at which a specific temperature occurs, Hana needs to find the inverse function. What is the inverse function d(T) that gives the depth in meters when the temperature is T degrees Celsius? Answer: d(T) = (3T - 15)/(12 - T) Solution: Start with the original function: T(d) = (12d + 15)/(d + 3) Replace T(d) with y: y = (12d + 15)/(d + 3) Swap the variables (d becomes y and y becomes d): d = (12y + 15)/(y + 3) Multiply both sides by (y + 3): d(y + 3) = 12y + 15 Distribute the d: dy + 3d = 12y + 15 Get all terms with y on one…
   Full step-by-step solution

   Step 1: Start with the original function: T(d) = (12d + 15)/(d + 3) Step 2: Replace T(d) with y: y = (12d + 15)/(d + 3) Step 3: Swap the variables (d becomes y and y becomes d): d = (12y + 15)/(y + 3) Step 4: Multiply both sides by (y + 3): d(y + 3) = 12y + 15 Step 5: Distribute the d: dy + 3d = 12y + 15 Step 6: Get all terms with y on one side: dy - 12y = 15 - 3d Step 7: Factor out y: y(d - 12) = 15 - 3d Step 8: Solve for y: y = (15 - 3d)/(d - 12) Step 9: Replace y with d(T) and d with T: d(T) = (15 - 3T)/(T - 12) Step 10: Multiply numerator and denominator by -1 to simplify: d(T) = (3T - 15)/(12 - T) The inverse function is d(T) = (3T - 15)/(12 - T).

2. A biomedical company is modeling the concentration of a new drug in a patient's bloodstream using the function C(t) = (2t + 3)/(t - 1), where C is the concentration in milligrams per liter and t is time in hours. To determine when the drug concentration reaches a specific level, the researchers need to find the inverse function. Find the inverse function t(C) that gives the time when the concentration equals C. Answer: t(C) = (C + 3)/(C - 2) Solution: C(t) = (2t + 3)/(t - 1) Our goal is to find the inverse function t(C). Replace C(t) with y for clarity in algebra.
   Full step-by-step solution

   We start with the function for concentration: C(t) = (2t + 3)/(t - 1) Our goal is to find the inverse function t(C). Step 1: Replace C(t) with y for clarity in algebra. Let y = (2t + 3)/(t - 1) Step 2: Swap t and y because for the inverse, we express t in terms of C (which is y here). So after swapping: t = (2y + 3)/(y - 1) Step 3: Solve for y in terms of t. Multiply both sides by (y - 1): t(y - 1) = 2y + 3 Step 4: Distribute t: t y - t = 2y + 3 Step 5: Bring terms with y to one side and other terms to the other side: t y - 2y = t + 3 Step 6: Factor y from the left side: y (t - 2) = t + 3 Step 7: Solve for y: y = (t + 3)/(t - 2) Step 8: Since we swapped t and y earlier, now y represents t(C) and t represents C. So we have: t(C) = (C + 3)/(C - 2) This is the inverse function. Final answer: t(C) = (C + 3)/(C - 2)

3. A function is represented graphically as a cubic curve with inflection point at (1, 2) and passing through points (0, 3) and (2, 1). The function has the form f(x) = a(x - h)³ + k. Find the algebraic expression for its inverse function f⁻¹(x). Answer: f⁻¹(x) = 1 + ∛(x - 2) Solution: Identify the vertex form: f(x) = a(x - 1)³ + 2 Use point (0, 3) to find 'a': 3 = a(0 - 1)³ + 2 3 = a(-1)³ + 2 3 = -a + 2 -a = 1, so a = -1 The function is f(x) = -(x - 1)³ + 2 To find the inverse, swap x and y: x = -(y - 1)³ + 2 Solve for y: x - 2 = -(y - 1)³ Multiply both sides by -1: 2 - x =…
   Full step-by-step solution

   Step 1: Identify the vertex form: f(x) = a(x - 1)³ + 2 Step 2: Use point (0, 3) to find 'a': 3 = a(0 - 1)³ + 2 Step 3: 3 = a(-1)³ + 2 Step 4: 3 = -a + 2 Step 5: -a = 1, so a = -1 Step 6: The function is f(x) = -(x - 1)³ + 2 Step 7: To find the inverse, swap x and y: x = -(y - 1)³ + 2 Step 8: Solve for y: x - 2 = -(y - 1)³ Step 9: Multiply both sides by -1: 2 - x = (y - 1)³ Step 10: Take cube root: ∛(2 - x) = y - 1 Step 11: Add 1 to both sides: y = 1 + ∛(2 - x) Step 12: Write in standard form: f⁻¹(x) = 1 + ∛(2 - x) The inverse function is f⁻¹(x) = 1 + ∛(2 - x)

4. A biomedical engineer is modeling the concentration of a new medication in a patient's bloodstream using the function C(t) = (5t + 3)/(2t - 7), where C represents concentration in milligrams per liter and t represents time in hours since administration. To determine when the concentration reaches a specific level, the engineer needs to find the inverse function. What is the inverse function t(C) that gives the time when the concentration is C? Answer: t(C) = (7C + 3)/(2C - 5) Solution: C(t) = (5t + 3)/(2t - 7) Replace C(t) with C for simplicity. C = (5t + 3)/(2t - 7) To find the inverse, we swap the roles of C and t. That means we solve for t in terms of C.
   Full step-by-step solution

   We start with the function: C(t) = (5t + 3)/(2t - 7) Step 1: Replace C(t) with C for simplicity. C = (5t + 3)/(2t - 7) Step 2: To find the inverse, we swap the roles of C and t. That means we solve for t in terms of C. So we have: C = (5t + 3)/(2t - 7) Step 3: Multiply both sides by (2t - 7) to eliminate the denominator. C(2t - 7) = 5t + 3 Step 4: Distribute C on the left side. 2Ct - 7C = 5t + 3 Step 5: Bring all terms with t to one side and other terms to the other side. 2Ct - 5t = 7C + 3 Step 6: Factor t from the left side. t(2C - 5) = 7C + 3 Step 7: Divide both sides by (2C - 5) to solve for t. t = (7C + 3)/(2C - 5) Step 8: This is the inverse function. We write it as: t(C) = (7C + 3)/(2C - 5) This matches the correct answer.

5. f(x) = (2x + 5)/(3x - 4), find f⁻¹(x) = ? Answer: (4x + 5)/(3x - 2) Solution: Replace f(x) with y: y = (2x + 5)/(3x - 4) Swap x and y: x = (2y + 5)/(3y - 4) Multiply both sides by (3y - 4): x(3y - 4) = 2y + 5 Distribute x: 3xy - 4x = 2y + 5 Move y terms to one side: 3xy - 2y = 4x + 5 Factor out y: y(3x - 2) = 4x + 5 Solve for y: y = (4x + 5)/(3x - 2) Replace y with…
   Full step-by-step solution

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

6. A function is represented graphically as a cubic curve with inflection point at (1, 2) and passing through points (0, 3) and (2, 1). The function has the form f(x) = ax³ + bx² + cx + d. Find the algebraic expression for the inverse function f⁻¹(x) at the point where f(x) = 2. Answer: 1 Solution: The problem states the function has an inflection point at (1, 2), which means f(1) = 2. For the inverse function f⁻¹(x), we know that f⁻¹(f(x)) = x.
   Full step-by-step solution

   Step 1: The problem states the function has an inflection point at (1, 2), which means f(1) = 2. Step 2: For the inverse function f⁻¹(x), we know that f⁻¹(f(x)) = x. Therefore, f⁻¹(f(1)) = f⁻¹(2) = 1. Step 3: Since f(1) = 2, then by definition of inverse functions, f⁻¹(2) must equal 1. Step 4: The answer is 1.

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

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