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Inverse Functions

Grade 12 · Algebra · Worksheet 2

  1. f(x) = (2x - 5)/(x + 3), find f⁻¹(x) = ? Answer: ______________
  2. A biomedical company is modeling the concentration of a new medication in a patient's bloodstream using the function C(t) = (2t + 3)/(t - 1), where C represents concentration in mg/L and t represents time in hours. To determine when the concentration reaches a specific level, they need to find the inverse function. What is the inverse function t(C) that gives the time when the concentration is C mg/L? Answer: ______________
  3. f(x) = (11x + 17)/(8x - 15). Find f⁻¹(x) = ? Answer: ______________
  4. Sophia is a materials scientist studying the thermal expansion of a new ceramic composite. The length L(t) of a sample in millimeters after t seconds of heating is given by the function L(t) = (7t + 9)/(t - 4), where t > 4. To predict how long it will take for the sample to reach a specific length, Sophia needs to find the inverse function that expresses time as a function of length. What is the inverse function t(L) that gives the time in seconds when the length is L millimeters? Answer: ______________
  5. Dr. Chen is analyzing the concentration of a medication in a patient's bloodstream over time. The concentration C(t) in milligrams per liter is modeled by the function C(t) = (3t + 2)/(t - 1), where t represents hours since administration. To determine when the concentration reaches a specific level, Dr. Chen needs to find the inverse function that gives time as a function of concentration. Find the inverse function t(C). Answer: ______________
  6. f(x) = (7x + 11)/(4x - 13), find f⁻¹(x) = ? Answer: ______________
  7. f(x) = (8x + 19)/(11x - 15). Find f⁻¹(x) = ? Answer: ______________
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Answer Key & Explanations

Inverse Functions · Grade 12 · Worksheet 2

  1. f(x) = (2x - 5)/(x + 3), find f⁻¹(x) = ? Answer: f⁻¹(x) = (-3x - 5)/(x - 2) Solution: To find the inverse function f⁻¹(x) for f(x) = (2x - 5)/(x + 3), follow these steps: y = (2x - 5)/(x + 3) This is the key step for finding an inverse function - we exchange the roles of x and y: x = (2y - 5)/(y + 3) We need to isolate y on one side of the equation.
    Full step-by-step solution

    To find the inverse function f⁻¹(x) for f(x) = (2x - 5)/(x + 3), follow these steps: Step 1: Replace f(x) with y y = (2x - 5)/(x + 3) Step 2: Swap x and y This is the key step for finding an inverse function - we exchange the roles of x and y: x = (2y - 5)/(y + 3) Step 3: Solve for y We need to isolate y on one side of the equation. First, multiply both sides by (y + 3) to eliminate the denominator: x(y + 3) = 2y - 5 Step 4: Distribute x on the left side: xy + 3x = 2y - 5 Step 5: Get all terms with y on one side and other terms on the other side: xy - 2y = -3x - 5 Step 6: Factor y out of the left side: y(x - 2) = -3x - 5 Step 7: Divide both sides by (x - 2) to solve for y: y = (-3x - 5)/(x - 2) Step 8: Write the final answer Since y now represents f⁻¹(x), we have: f⁻¹(x) = (-3x - 5)/(x - 2) This is the inverse function of f(x).

  2. A biomedical company is modeling the concentration of a new medication in a patient's bloodstream using the function C(t) = (2t + 3)/(t - 1), where C represents concentration in mg/L and t represents time in hours. To determine when the concentration reaches a specific level, they need to find the inverse function. What is the inverse function t(C) that gives the time when the concentration is C mg/L? Answer: t(C) = (C + 3)/(C - 2) Solution: C(t) = (2t + 3)/(t - 1) We want to find the inverse function t(C), so we need to solve for t in terms of C. Replace C(t) with C. C = (2t + 3)/(t - 1) Multiply both sides by (t - 1) to eliminate the denominator.
    Full step-by-step solution

    We start with the concentration function: C(t) = (2t + 3)/(t - 1) We want to find the inverse function t(C), so we need to solve for t in terms of C. Step 1: Replace C(t) with C. C = (2t + 3)/(t - 1) Step 2: Multiply both sides by (t - 1) to eliminate the denominator. C(t - 1) = 2t + 3 Step 3: Distribute C on the left side. Ct - C = 2t + 3 Step 4: Bring all terms with t to one side and constant terms to the other side. Ct - 2t = C + 3 Step 5: Factor t from the left side. t(C - 2) = C + 3 Step 6: Solve for t by dividing both sides by (C - 2). t = (C + 3)/(C - 2) Step 7: Write the inverse function notation. t(C) = (C + 3)/(C - 2) This is the inverse function that gives the time t when the concentration is C mg/L.

  3. f(x) = (11x + 17)/(8x - 15). Find f⁻¹(x) = ? Answer: (15x + 17)/(8x - 11) Solution: Replace f(x) with y: y = (11x + 17)/(8x - 15) Swap x and y: x = (11y + 17)/(8y - 15) Multiply both sides by (8y - 15): x(8y - 15) = 11y + 17 Distribute x: 8xy - 15x = 11y + 17 Move all y terms to the left, constants to the right: 8xy - 11y = 15x + 17 Factor out y: y(8x - 11) = 15x + 17 Solve for…
    Full step-by-step solution

    Step 1: Replace f(x) with y: y = (11x + 17)/(8x - 15) Step 2: Swap x and y: x = (11y + 17)/(8y - 15) Step 3: Multiply both sides by (8y - 15): x(8y - 15) = 11y + 17 Step 4: Distribute x: 8xy - 15x = 11y + 17 Step 5: Move all y terms to the left, constants to the right: 8xy - 11y = 15x + 17 Step 6: Factor out y: y(8x - 11) = 15x + 17 Step 7: Solve for y: y = (15x + 17)/(8x - 11) Step 8: Replace y with f⁻¹(x): f⁻¹(x) = (15x + 17)/(8x - 11)

  4. Sophia is a materials scientist studying the thermal expansion of a new ceramic composite. The length L(t) of a sample in millimeters after t seconds of heating is given by the function L(t) = (7t + 9)/(t - 4), where t > 4. To predict how long it will take for the sample to reach a specific length, Sophia needs to find the inverse function that expresses time as a function of length. What is the inverse function t(L) that gives the time in seconds when the length is L millimeters? Answer: t(L) = (4L + 9)/(L - 7) Solution: Write the original function with y instead of L(t): y = (7t + 9)/(t - 4) Swap the variables t and y: t = (7y + 9)/(y - 4) Multiply both sides by (y - 4): t(y - 4) = 7y + 9 Distribute t on the left: ty - 4t = 7y + 9 Move all terms with y to the left, and other terms to the right: ty - 7y = 4t + 9…
    Full step-by-step solution

    Step 1: Write the original function with y instead of L(t): y = (7t + 9)/(t - 4) Step 2: Swap the variables t and y: t = (7y + 9)/(y - 4) Step 3: Multiply both sides by (y - 4): t(y - 4) = 7y + 9 Step 4: Distribute t on the left: ty - 4t = 7y + 9 Step 5: Move all terms with y to the left, and other terms to the right: ty - 7y = 4t + 9 Step 6: Factor out y on the left: y(t - 7) = 4t + 9 Step 7: Solve for y: y = (4t + 9)/(t - 7) Step 8: Replace y with t(L) and t with L: t(L) = (4L + 9)/(L - 7) The inverse function is t(L) = (4L + 9)/(L - 7).

  5. Dr. Chen is analyzing the concentration of a medication in a patient's bloodstream over time. The concentration C(t) in milligrams per liter is modeled by the function C(t) = (3t + 2)/(t - 1), where t represents hours since administration. To determine when the concentration reaches a specific level, Dr. Chen needs to find the inverse function that gives time as a function of concentration. Find the inverse function t(C). Answer: t(C) = (C + 2)/(C - 3) Solution: C(t) = (3t + 2)/(t - 1) Replace C(t) with C for simplicity. C = (3t + 2)/(t - 1) Swap the roles of C and t because we are finding the inverse. This means we solve for t in terms of C.
    Full step-by-step solution

    We start with the function: C(t) = (3t + 2)/(t - 1) Step 1: Replace C(t) with C for simplicity. C = (3t + 2)/(t - 1) Step 2: Swap the roles of C and t because we are finding the inverse. This means we solve for t in terms of C. So we have: C = (3t + 2)/(t - 1) Step 3: Multiply both sides by (t - 1) to eliminate the denominator. C(t - 1) = 3t + 2 Step 4: Distribute C on the left side. Ct - C = 3t + 2 Step 5: Bring all terms containing t to one side and other terms to the other side. Ct - 3t = C + 2 Step 6: Factor t from the left side. t(C - 3) = C + 2 Step 7: Solve for t by dividing both sides by (C - 3). t = (C + 2)/(C - 3) Step 8: Write the inverse function notation. t(C) = (C + 2)/(C - 3) This is the inverse function, giving time t in hours as a function of concentration C.

  6. f(x) = (7x + 11)/(4x - 13), find f⁻¹(x) = ? Answer: (13x + 11)/(4x - 7) Solution: Replace f(x) with y: y = (7x + 11)/(4x - 13) Swap x and y: x = (7y + 11)/(4y - 13) Multiply both sides by (4y - 13): x(4y - 13) = 7y + 11 Distribute x: 4xy - 13x = 7y + 11 Move all y terms to one side: 4xy - 7y = 13x + 11 Factor out y: y(4x - 7) = 13x + 11 Solve for y: y = (13x + 11)/(4x - 7)…
    Full step-by-step solution

    Step 1: Replace f(x) with y: y = (7x + 11)/(4x - 13) Step 2: Swap x and y: x = (7y + 11)/(4y - 13) Step 3: Multiply both sides by (4y - 13): x(4y - 13) = 7y + 11 Step 4: Distribute x: 4xy - 13x = 7y + 11 Step 5: Move all y terms to one side: 4xy - 7y = 13x + 11 Step 6: Factor out y: y(4x - 7) = 13x + 11 Step 7: Solve for y: y = (13x + 11)/(4x - 7) Step 8: Replace y with f⁻¹(x): f⁻¹(x) = (13x + 11)/(4x - 7)

  7. f(x) = (8x + 19)/(11x - 15). Find f⁻¹(x) = ? Answer: (15x + 19)/(11x - 8) Solution: Replace f(x) with y: y = (8x + 19)/(11x - 15) Swap x and y: x = (8y + 19)/(11y - 15) Multiply both sides by (11y - 15): x(11y - 15) = 8y + 19 Distribute x: 11xy - 15x = 8y + 19 Move all y terms to the left, constants to the right: 11xy - 8y = 15x + 19 Factor out y: y(11x - 8) = 15x + 19 Solve…
    Full step-by-step solution

    Step 1: Replace f(x) with y: y = (8x + 19)/(11x - 15) Step 2: Swap x and y: x = (8y + 19)/(11y - 15) Step 3: Multiply both sides by (11y - 15): x(11y - 15) = 8y + 19 Step 4: Distribute x: 11xy - 15x = 8y + 19 Step 5: Move all y terms to the left, constants to the right: 11xy - 8y = 15x + 19 Step 6: Factor out y: y(11x - 8) = 15x + 19 Step 7: Solve for y: y = (15x + 19)/(11x - 8) Step 8: Replace y with f⁻¹(x): f⁻¹(x) = (15x + 19)/(11x - 8)