Question 1: Find the inverse function of f(x) = 3x + 5.

Answer: f^{-1}(x) = (x - 5)/3

Question 2: Verify whether g(x) = (x - 2)/4 is the inverse of h(x) = 4x + 2. Show your working.

Answer: Yes, because g(h(x)) = x and h(g(x)) = x

Question 3: If a function f(x) = (2x - 1)/3, what is its inverse? Write your answer clearly.

Answer: f^{-1}(x) = (3x + 1)/2

Question 4: A student claims that the inverse of f(x) = x^2 is f^{-1}(x) = √x. Explain why this is incorrect and provide the correct inverse.

Answer: Because f(x) = x^2 is not one-to-one over all real numbers, its inverse is f^{-1}(x) = |x|, but when restricted to x ≥ 0, the inverse is √x.

Question 5: Plot the graph of y = (x + 1)/2 and its inverse on the grid provided.

Answer: Student plots the lines y = (x + 1)/2 and y = 2x - 1 correctly.

Question 6: A function is given as f(x) = 2x + 3. Its inverse is incorrectly written as f^{-1}(x) = (x - 3)/2. Identify and correct the mistake.

Answer: There is no mistake; the inverse is correctly written as f^{-1}(x) = (x - 3)/2.

Question 7: Construct a function f(x) such that its inverse is f^{-1}(x) = (x + 4)/5. Write the original function.

Answer: f(x) = 5x - 4

Question 8: A student claims that the inverse of f(x) = x^3 is f^{-1}(x) = √x. Explain whether this is correct and, if not, find the correct inverse.

Answer: Incorrect. The inverse of f(x) = x^3 is f^{-1}(x) = ∛x.

Question 9: Given the function f(x) = (x - 2)/5, verify that its inverse is f^{-1}(x) = 5x + 2. Show your working.

Answer: f(f^{-1}(x)) = (5x + 2 - 2)/5 = x, and f^{-1}(f(x)) = 5((x - 2)/5) + 2 = x

Question 10: A function f(x) = 4x + 7 is given. Find its inverse and verify by substitution.

Answer: Inverse: f^{-1}(x) = (x - 7)/4; verification: f(f^{-1}(x)) = x, f^{-1}(f(x)) = x

Question 11: Identify the mistake in the following inverse verification: For f(x) = 2x + 3, they claimed the inverse is f^{-1}(x) = (x + 3)/2, but when checking, f(f^{-1}(x)) = 2((x + 3)/2) + 3 = x + 6, which is not x. Correct the inverse function.

Answer: The inverse is f^{-1}(x) = (x - 3)/2; the mistake was in the sign used in the inverse formula.