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

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

Question 2: Verify if g(x) = (x + 5)/3 is the inverse of f(x) = 3x + 5.

Answer: Yes, because composing f and g gives the identity function.

Question 3: Given h(x) = (2x - 4)/5, find its inverse.

Answer: h^{-1}(x) = (5x + 4)/2

Question 4: Construct the inverse of the function y = (x - 2)/4 and verify your answer.

Answer: Inverse: y = 4x + 2; Verification: composition gives identity.

Question 5: A function is defined as f(x) = 2x + 7. Find its inverse and explain your steps.

Answer: Inverse: f^{-1}(x) = (x - 7)/2; achieved by swapping x and y and solving.

Question 6: Determine if the function f(x) = x^3 + 1 has an inverse over all real numbers. Justify your answer.

Answer: Yes, because it is strictly increasing over all real numbers, so it has an inverse.

Question 7: Find the inverse of y = (3x - 2)/(x + 4).

Answer: Inverse: x = (2y + 8)/(3 - y)

Question 8: Verify whether f(x) = (x - 1)/(x + 2) and g(x) = (2x + 1)/(x - 3) are inverses.

Answer: Compose f(g(x)) and g(f(x)), check if both simplify to x.

Question 9: A real-world scenario: A person’s height in meters is modeled by h(x) = 2x + 1, where x is age in years. Find the inverse to determine age from height.

Answer: Age: x = (height - 1)/2

Question 10: Plot the graph of y = 2x and its inverse on the grid. Describe what you observe.

Answer: The graph of y=2x is symmetric about y=x, illustrating inverse relationship.

Question 11: Identify a common mistake students make when finding inverses and explain how to correct it.

Answer: Swapping variables incorrectly or not solving for y properly.

Question 12: Challenge: The function f(x) = (x^2 + 1)/(x - 2) is not one-to-one over all real numbers. Find its inverse on the restricted domain x > 2.

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