Question 1: Find the inverse of the function f(x) = 3x + 4.

Answer: f^{-1}(x) = (x - 4)/3

Question 2: Verify if g(x) = (x - 4)/3 is the inverse of f(x) = 3x + 4. Show your working.

Answer: Compose f and g: f(g(x)) = 3((x - 4)/3)+4 = x - 4 + 4 = x; g(f(x)) = ((3x + 4) - 4)/3 = 3x/3 = x. Both compositions equal x, so g(x) is the inverse.

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

Answer: f^{-1}(x) = (3x + 5)/2

Question 4: Construct a function g(x) such that g is the inverse of f(x) = (2x - 5)/3. Verify your inverse by composition.

Answer: g(x) = (3x + 5)/2; verify: f(g(x)) = (2((3x + 5)/2) - 5)/3 = (3x + 5 - 5)/3 = 3x/3 = x; g(f(x)) = (3((2x - 5)/3) + 5)/2 = (2x - 5 + 5)/2 = 2x/2 = x.

Question 5: Determine if the function h(x) = 2x + 1 is invertible. If yes, find its inverse.

Answer: Yes; inverse h^{-1}(x) = (x - 1)/2

Question 6: Plot the graph of y = 2x and its inverse y = x/2 on the grid. Do the graphs appear to be reflections across the line y = x?

Answer: Yes, they are reflections across y = x.

Question 7: Given the function f(x) = x^3, find its inverse and verify it.

Answer: Inverse is f^{-1}(x) = ∛x; verification: f(f^{-1}(x)) = (∛x)^3 = x; f^{-1}(f(x)) = ∛(x^3) = x.

Question 8: A function f(x) is given by f(x) = (x + 2)/(x - 3). Find and verify its inverse.

Answer: Inverse: f^{-1}(x) = (3x + 2)/(1 - x); verification involves composing and simplifying both ways.

Question 9: Identify the common mistake when students forget to swap x and y when finding an inverse and correct the mistake with an example.

Answer: Common mistake: Solving directly for y without swapping variables. Example correction: For f(x) = 2x + 3, swap to get y = 2x + 3, then solve for x to find inverse.

Question 10: Challenge: Find the inverse of f(x) = (x^2 + 4) for x ≥ 0, and explain why the inverse is restricted to x ≥ 4.

Answer: Inverse: f^{-1}(x) = √(x - 4); domain restriction ensures the inverse function is a proper function (one-to-one).