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

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

Question 2: Verify whether the function f(x) = (2x - 4)/3 is invertible by calculating f(f^{-1}(x)).

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

Question 3: Construct the inverse function of f(x) = (x + 7)/2.

Answer: f^{-1}(x) = 2x - 7

Question 4: Plot the graph of y = 2x on the provided grid. Then, draw its inverse.

Answer: Student draws y=2x and its reflection across y=x

Question 5: Given the function f(x) = x^3 + 2, find its inverse.

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

Question 6: Determine if the function f(x) = x^2 + 1 is invertible over all real numbers. Explain your reasoning.

Answer: No, because it is not one-to-one over all reals; it fails the horizontal line test.

Question 7: Calculate the inverse of f(x) = (4x - 1)/5 and verify it by composition.

Answer: Inverse: f^{-1}(x) = (5x + 1)/4; verification confirms f(f^{-1}(x))=x and f^{-1}(f(x))=x

Question 8: A function is defined as f(x) = 2x + 3. Its inverse is mistakenly written as f^{-1}(x) = (x - 3)/2. Identify and explain the error.

Answer: Correct inverse: f^{-1}(x) = (x - 3)/2; no error, the student's inverse is correct.

Question 9: Construct a function that is not invertible and explain why.

Answer: f(x) = x^2 (over all reals) is not invertible because it is not one-to-one.

Question 10: Find the inverse of f(x) = (x + 2)^2, where x ≥ -2.

Answer: f^{-1}(x) = √x - 2

Question 11: Explain how verifying an inverse function helps confirm that two functions are inverses.

Answer: Verification by composition (f(f^{-1}(x))=x and vice versa) confirms the functions are inverses.