Inverse Functions

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Answer Key & Explanations

Inverse Functions · Grade 12 · Worksheet 3

1. A graph on a coordinate plane shows a curve that is symmetric about the line y = x. The function f(x) passes through the points (1, -3), (3, 1), and (5, 5). Given that f(x) = ax^3 + bx^2 + cx + d, find f⁻¹(5). Answer: 5 Solution: The graph of f(x) is symmetric about the line y = x, which means the function is its own inverse. For a function and its inverse, if (a, b) lies on f(x), then (b, a) lies on f⁻¹(x).
   Full step-by-step solution

   Step 1: The graph of f(x) is symmetric about the line y = x, which means the function is its own inverse. For a function and its inverse, if (a, b) lies on f(x), then (b, a) lies on f⁻¹(x). Step 2: We are given points on f(x): (1, -3), (3, 1), (5, 5). Step 3: To find f⁻¹(5), we need to find the x-value such that f(x) = 5. From the given points, we see that (5, 5) is on f(x), meaning f(5) = 5. Step 4: By the definition of inverse functions, f⁻¹(5) = the x-value where f(x) = 5. Since f(5) = 5, we have f⁻¹(5) = 5. The answer is 5.

2. A pharmaceutical 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 after administration. The researchers need to determine at what time the concentration reaches a specific level. Find the inverse function C⁻¹(c) that would allow them to calculate the time when the concentration equals any given value c. Answer: C⁻¹(c) = (c + 3)/(c - 2) Solution: We are given the function C(t) = (2t + 3)/(t - 1) and need to find the inverse function C⁻¹(c). Write the function with y = C(t) and t as the input variable.
   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(c). Step 1: Write the function with y = C(t) and t as the input variable. y = (2t + 3)/(t - 1) Step 2: Swap the roles of t and y because for the inverse, we want time as a function of concentration. So let t = C⁻¹(y), but in the inverse function notation, we usually write it as C⁻¹(c) where c is the concentration. So after swapping: t = (2y + 3)/(y - 1) Here, y now represents the time in the original function, but after swapping, t is concentration and y is time. Let's avoid confusion by renaming variables carefully. Better approach: Let c = C(t) = (2t + 3)/(t - 1). We want t in terms of c. Step 3: Start with c = (2t + 3)/(t - 1). Step 4: Multiply both sides by (t - 1) to eliminate the denominator: c(t - 1) = 2t + 3. Step 5: Expand the left side: c t - c = 2t + 3. Step 6: Bring all terms containing t to one side and other terms to the other side: c t - 2t = c + 3. Step 7: Factor t from the left side: t(c - 2) = c + 3. Step 8: Solve for t: t = (c + 3)/(c - 2). Step 9: This t is the time at which concentration equals c, so this is the inverse function: C⁻¹(c) = (c + 3)/(c - 2). Final answer: C⁻¹(c) = (c + 3)/(c - 2).

3. Mere is an agricultural scientist modeling the rate of water absorption by a special type of soil. She uses the function A(t) = (3t + 13)/(t - 5), where A represents the absorption rate in liters per hour and t represents time in hours since the start of an irrigation experiment. To predict when the absorption rate will reach a specific target level, Mere needs to find the inverse function. What is the inverse function A⁻¹(x) that gives the time when the absorption rate is x liters per hour? Answer: A⁻¹(x) = (5x + 13)/(x - 3) Solution: Start with the original function: A(t) = (3t + 13)/(t - 5) Replace A(t) with y: y = (3t + 13)/(t - 5) Swap the variables (t becomes x and y becomes t): x = (3y + 13)/(y - 5) Multiply both sides by (y - 5) to clear the denominator: x(y - 5) = 3y + 13 Distribute x on the left side: xy - 5x = 3y +…
   Full step-by-step solution

   Step 1: Start with the original function: A(t) = (3t + 13)/(t - 5) Step 2: Replace A(t) with y: y = (3t + 13)/(t - 5) Step 3: Swap the variables (t becomes x and y becomes t): x = (3y + 13)/(y - 5) Step 4: Multiply both sides by (y - 5) to clear the denominator: x(y - 5) = 3y + 13 Step 5: Distribute x on the left side: xy - 5x = 3y + 13 Step 6: Bring all terms with y to one side and constants to the other: xy - 3y = 5x + 13 Step 7: Factor out y from the left side: y(x - 3) = 5x + 13 Step 8: Solve for y by dividing both sides by (x - 3): y = (5x + 13)/(x - 3) Step 9: Replace y with A⁻¹(x): A⁻¹(x) = (5x + 13)/(x - 3) The inverse function is A⁻¹(x) = (5x + 13)/(x - 3).

4. A function is represented graphically as a parabola opening upward with vertex at (2, -3) and passing through point (4, 1). If this function is defined as f(x) = a(x - h)² + k, find the algebraic expression for its inverse function f⁻¹(x). Answer: f⁻¹(x) = 2 ± √((x + 3)/1) Solution: f(x) = a(x - h)² + k Vertex = (2, -3) → so h = 2, k = -3.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Identify the given function form and parameters** The problem says: f(x) = a(x - h)² + k Vertex = (2, -3) → so h = 2, k = -3. Thus: f(x) = a(x - 2)² - 3 --- **Step 2: Use the given point to find a** Point (4, 1) is on the parabola: 1 = a(4 - 2)² - 3 1 = a(2)² - 3 1 = 4a - 3 4a = 4 a = 1 So the function is: f(x) = 1*(x - 2)² - 3 f(x) = (x - 2)² - 3 --- **Step 3: Write the equation for the inverse function** To find the inverse, swap x and y in y = (x - 2)² - 3: x = (y - 2)² - 3 --- **Step 4: Solve for y** x + 3 = (y - 2)² (y - 2)² = x + 3 y - 2 = ±√(x + 3) y = 2 ± √(x + 3) --- **Step 5: Write the inverse function notation** The inverse is: f⁻¹(x) = 2 ± √(x + 3) Since a = 1, we can write √((x + 3)/1) instead of √(x + 3) if we want to match the given answer format exactly. So: f⁻¹(x) = 2 ± √((x + 3)/1) --- **Final answer:** f⁻¹(x) = 2 ± √((x + 3)/1)

5. A function is represented graphically as a parabola opening upward with vertex at (2, -3) and passing through point (4, 1). If this function has an inverse over a restricted domain, what is the equation of the inverse function? Answer: y = 2 + √(x + 3) Solution: - Vertex at (2, -3) - Passes through (4, 1) y = a(x - h)^2 + k, where (h, k) is the vertex.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Find the equation of the parabola** We are told: - Vertex at (2, -3) - Opens upward - Passes through (4, 1) Vertex form of a parabola: y = a(x - h)^2 + k, where (h, k) is the vertex. So: y = a(x - 2)^2 - 3 Plug in (4, 1): 1 = a(4 - 2)^2 - 3 1 = a(2)^2 - 3 1 = 4a - 3 4a = 4 a = 1 Thus the equation is: y = (x - 2)^2 - 3 --- **Step 2: Determine the restricted domain for an inverse** The parabola opens upward, vertex at x = 2. To have an inverse, we restrict the domain so the function is one-to-one. Two possibilities: x ≥ 2 (right branch) or x ≤ 2 (left branch). Usually, we take the right branch (x ≥ 2) so that the inverse will be a function. --- **Step 3: Find the inverse function** We start with: y = (x - 2)^2 - 3, with domain x ≥ 2. Switch x and y to find the inverse: x = (y - 2)^2 - 3 Solve for y: x + 3 = (y - 2)^2 Take square root: y - 2 = ±√(x + 3) --- **Step 4: Choose the correct sign** Original domain: x ≥ 2, so in the original function, x - 2 ≥ 0. When we swap variables, the range of the original becomes the domain of the inverse, and the original domain x ≥ 2 becomes the condition for choosing the sign. Original: y = (x - 2)^2 - 3, x ≥ 2. Then y ≥ -3 (since vertex is minimum at -3 for x ≥ 2 actually wait — careful: For x ≥ 2, y increases from -3 upward. So range of original: y ≥ -3.) For inverse: domain x ≥ -3, range y ≥ 2 (since original domain x ≥ 2 becomes range of inverse y ≥ 2). Thus we choose the positive square root: y - 2 = +√(x + 3) So: y = 2 + √(x + 3) --- **Final answer:** y = 2 + √(x + 3)

6. f(x) = (17x + 23)/(14x - 19). Find f⁻¹(x) = ? Answer: (19x + 23)/(14x - 17) Solution: Replace f(x) with y: y = (17x + 23)/(14x - 19) Swap x and y: x = (17y + 23)/(14y - 19) Multiply both sides by (14y - 19): x(14y - 19) = 17y + 23 Distribute x: 14xy - 19x = 17y + 23 Move all y terms to the left, constants to the right: 14xy - 17y = 19x + 23 Factor out y: y(14x - 17) = 19x + 23…
   Full step-by-step solution

   Step 1: Replace f(x) with y: y = (17x + 23)/(14x - 19) Step 2: Swap x and y: x = (17y + 23)/(14y - 19) Step 3: Multiply both sides by (14y - 19): x(14y - 19) = 17y + 23 Step 4: Distribute x: 14xy - 19x = 17y + 23 Step 5: Move all y terms to the left, constants to the right: 14xy - 17y = 19x + 23 Step 6: Factor out y: y(14x - 17) = 19x + 23 Step 7: Solve for y: y = (19x + 23)/(14x - 17) Step 8: Replace y with f⁻¹(x): f⁻¹(x) = (19x + 23)/(14x - 17)

7. 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).