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

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

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

Answer: Yes, because g(f(x)) = (2(3x+4)-4)/3 = (6x+8-4)/3 = (6x+4)/3 = 2x+4/3; g(f(x)) simplifies to x, confirming inverse.

Question 3: Construct the inverse function for f(x) = (x - 2)/5.

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

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

Answer: Students draw the line y=2x and its reflection y=(1/2)x across y=x.

Question 5: Determine whether the function h(x) = 4x - 7 is invertible. Justify your answer.

Answer: Yes, because it's a linear function with a non-zero coefficient, so it is one-to-one and invertible.

Question 6: Given f(x) = (x+3)/2, find f^{-1}(x) and verify that it is the inverse.

Answer: f^{-1}(x) = 2x - 3; verify by showing f(f^{-1}(x)) = x and f^{-1}(f(x)) = x.

Question 7: A function f(x) = x^3 is given. Is it invertible? Explain your reasoning.

Answer: Yes, because x^3 is a one-to-one function over all real numbers.

Question 8: Construct a real-world scenario where finding an inverse function would be useful, and briefly explain how you would verify it.

Answer: Example: Converting Celsius to Fahrenheit and finding the inverse to convert back. Verify by confirming the composition yields the original temperature.

Question 9: Identify and correct the mistake in the following inverse verification: g(x) = (x+4)/3; student claims g(f(x)) = (3x + 4 + 4)/3 = (3x + 8)/3 = x + 8/3.

Answer: Correct step: g(f(x)) = (3x + 4 + 4)/3 = (3x + 8)/3; the mistake is the assumption that the composition simplifies to x, but actually, it simplifies to x + 8/3. The functions are not inverse.

Question 10: Challenge: Find a function f(x) such that its inverse is f^{-1}(x) = (x - 2)/3. Write the original function.

Answer: f(x) = 3x + 2