Inverse Functions

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Inverse Functions · Grade 12 · Worksheet 2

1. Aroha is a climate scientist studying the relationship between ocean temperature and depth in a specific trench. The temperature T(d) in degrees Celsius at a depth d kilometers is modeled by the rational function T(d) = (9d + 7)/(d - 3), where d > 3. To predict the depth at which a specific temperature occurs, Aroha needs to find the inverse function. Determine the inverse function d(T) that gives the depth in kilometers when the temperature is T degrees Celsius. Answer: d(T) = (3T + 7)/(T - 9) Solution: Write the original function with y instead of T(d): y = (9d + 7)/(d - 3) Swap the variables d and y: d = (9y + 7)/(y - 3) Multiply both sides by (y - 3) to clear the denominator: d(y - 3) = 9y + 7 Distribute d on the left side: dy - 3d = 9y + 7 Bring all terms with y to one side and constant…
   Full step-by-step solution

   Step 1: Write the original function with y instead of T(d): y = (9d + 7)/(d - 3) Step 2: Swap the variables d and y: d = (9y + 7)/(y - 3) Step 3: Multiply both sides by (y - 3) to clear the denominator: d(y - 3) = 9y + 7 Step 4: Distribute d on the left side: dy - 3d = 9y + 7 Step 5: Bring all terms with y to one side and constant terms to the other: dy - 9y = 3d + 7 Step 6: Factor y out of the left side: y(d - 9) = 3d + 7 Step 7: Solve for y by dividing both sides by (d - 9): y = (3d + 7)/(d - 9) Step 8: Replace y with d(T) and the variable d with T: d(T) = (3T + 7)/(T - 9) The inverse function is d(T) = (3T + 7)/(T - 9).

2. f(x) = (13x + 9)/(11x - 8). Find f⁻¹(x) = ? Answer: (8x + 9)/(11x - 13) Solution: Replace f(x) with y: y = (13x + 9)/(11x - 8) Swap x and y: x = (13y + 9)/(11y - 8) Multiply both sides by (11y - 8): x(11y - 8) = 13y + 9 Distribute x: 11xy - 8x = 13y + 9 Move all y terms to the left, constants to the right: 11xy - 13y = 8x + 9 Factor out y: y(11x - 13) = 8x + 9 Solve for y: y…
   Full step-by-step solution

   Step 1: Replace f(x) with y: y = (13x + 9)/(11x - 8) Step 2: Swap x and y: x = (13y + 9)/(11y - 8) Step 3: Multiply both sides by (11y - 8): x(11y - 8) = 13y + 9 Step 4: Distribute x: 11xy - 8x = 13y + 9 Step 5: Move all y terms to the left, constants to the right: 11xy - 13y = 8x + 9 Step 6: Factor out y: y(11x - 13) = 8x + 9 Step 7: Solve for y: y = (8x + 9)/(11x - 13) Step 8: Replace y with f⁻¹(x): f⁻¹(x) = (8x + 9)/(11x - 13)

3. Olivia is a materials scientist studying the thermal expansion of a new ceramic composite. The length L(t) of a sample in millimeters at temperature t (in degrees Celsius) is modeled by the function L(t) = (3t + 1) / (t - 3). To predict the temperature required to achieve a specific length for manufacturing tolerances, Olivia needs the inverse function that expresses temperature as a function of length. Find the inverse function t(L) that gives the temperature in degrees Celsius when the length is L millimeters. Answer: t(L) = (3L + 1) / (L - 3) Solution: Write the original function with y instead of L(t): y = (3t + 1)/(t - 3) Swap the variables t and y: t = (3y + 1)/(y - 3) Multiply both sides by (y - 3) to eliminate the denominator: t(y - 3) = 3y + 1 Distribute t: ty - 3t = 3y + 1 Bring all terms with y to one side: ty - 3y = 3t + 1 Factor out…
   Full step-by-step solution

   Step 1: Write the original function with y instead of L(t): y = (3t + 1)/(t - 3) Step 2: Swap the variables t and y: t = (3y + 1)/(y - 3) Step 3: Multiply both sides by (y - 3) to eliminate the denominator: t(y - 3) = 3y + 1 Step 4: Distribute t: ty - 3t = 3y + 1 Step 5: Bring all terms with y to one side: ty - 3y = 3t + 1 Step 6: Factor out y: y(t - 3) = 3t + 1 Step 7: Divide both sides by (t - 3): y = (3t + 1)/(t - 3) Step 8: Replace y with t(L): t(L) = (3L + 1)/(L - 3) The inverse function is t(L) = (3L + 1)/(L - 3).

4. 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 the function M(t) = 50e^(-0.00012t). To determine the age of an archaeological sample when its mass is measured, 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) = (-ln(M/50))/0.00012 Solution: Start with the original function: M = 50e^(-0.00012t) Divide both sides by 50: M/50 = e^(-0.00012t) Take the natural logarithm of both sides: ln(M/50) = ln(e^(-0.00012t)) Simplify using logarithm properties: ln(M/50) = -0.00012t Solve for t: t = ln(M/50)/(-0.00012) Simplify the expression: t =…
   Full step-by-step solution

   Step 1: Start with the original function: M = 50e^(-0.00012t) Step 2: Divide both sides by 50: M/50 = e^(-0.00012t) Step 3: Take the natural logarithm of both sides: ln(M/50) = ln(e^(-0.00012t)) Step 4: Simplify using logarithm properties: ln(M/50) = -0.00012t Step 5: Solve for t: t = ln(M/50)/(-0.00012) Step 6: Simplify the expression: t = (-ln(M/50))/0.00012 Therefore, the inverse function is t(M) = (-ln(M/50))/0.00012

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

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

6. 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 is the concentration in milligrams per liter and t is time in hours since administration. To determine when the concentration reaches a specific level, the researchers need to find the inverse function. What is the inverse function C⁻¹(x) that gives the time when the concentration is x mg/L? Answer: C⁻¹(x) = (x + 3)/(x - 2) Solution: We are given the function C(t) = (2t + 3)/(t - 1) and need to find the inverse function C⁻¹(x). Write the function with y in terms of t. Let y = C(t), so: y = (2t + 3)/(t - 1) Swap variables.
   Full step-by-step solution

   We are given the function C(t) = (2t + 3)/(t - 1) and need to find the inverse function C⁻¹(x). Step 1: Write the function with y in terms of t. Let y = C(t), so: y = (2t + 3)/(t - 1) Step 2: Swap variables. For the inverse, we swap t and y: t = (2y + 3)/(y - 1) Here, t is now the output (time) and y is the input (concentration). But usually we write the inverse as C⁻¹(x) = something, so let's rename: Let x = concentration, then C⁻¹(x) = time t. So after swapping: t = (2x + 3)/(x - 1) But careful: originally y = (2t + 3)/(t - 1). After swapping, we get: t = (2y + 3)/(y - 1) Now rename y as x (concentration), and t as C⁻¹(x). So: C⁻¹(x) = (2x + 3)/(x - 1) ??? That’s not the given answer yet — let's solve properly. Step 3: Solve for the swapped variable. We start with: x = (2t + 3)/(t - 1) Multiply both sides by (t - 1): x(t - 1) = 2t + 3 xt - x = 2t + 3 Step 4: Bring terms with t to one side. xt - 2t = x + 3 Factor t: t(x - 2) = x + 3 Step 5: Solve for t. t = (x + 3)/(x - 2) Step 6: Write the inverse function. C⁻¹(x) = (x + 3)/(x - 2) This matches the given correct answer. Explanation: We swapped input and output because the inverse function reverses the roles. Then we solved algebraically for the original input variable (time t) in terms of the original output variable (concentration x). The domain of the inverse will be x ≠ 2 (since denominator x - 2 ≠ 0), which corresponds to the range of the original function excluding 2.