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

Answer: f⁻¹(x) = (x - 5) / 3

Question 2: Calculate f⁻¹(14) when f(x) = 2x - 4.

Answer: 9

Question 3: Plot the graph of y = 2x on the grid. Then, determine the inverse function f⁻¹(x).

Answer: f⁻¹(x) = x/2

Question 4: Construct a triangle on the grid with sides of length 3, 4, and 5, and label the coordinates of the vertices. (This is to practice plotting inverse functions.)

Answer:

Question 5: Solve for x: f(x) = 7 when f(x) = 4x + 1.

Answer: 2

Question 6: The function f(x) = (x + 2) / 3 has an inverse. Find f⁻¹(x).

Answer: f⁻¹(x) = 3x - 2

Question 7: A real-world scenario: A car rental company charges a fixed fee plus a rate per hour. The total cost in pounds is given by C(h) = 20 + 15h, where h is hours rented. Find the inverse function, expressing hours in terms of cost.

Answer: h = (C - 20) / 15

Question 8: If f(x) = -2x + 8, what is f⁻¹(0)?

Answer: 4

Question 9: Identify and correct the mistake in the following inverse function: given f(x) = 5x - 3, the inverse is written as f⁻¹(x) = (x + 3) / 5.

Answer: Incorrect; should be f⁻¹(x) = (x + 3) / 5

Question 10: A function g(x) is given by g(x) = x². Is g invertible over all real numbers? If not, specify the domain where it is invertible and find its inverse there.

Answer: g(x) = x² is invertible over x ≥ 0; inverse is g⁻¹(x) = √x.

Question 11: Explain a common mistake students make when finding the inverse of a linear function and how to avoid it.

Answer: Common mistake: swapping x and y without solving for y. To avoid, always solve the swapped equation for y.

Question 12: Given the function f(x) = 2x + 1, verify that the inverse function you found is correct by showing that f(f⁻¹(x)) = x.

Answer: f(f⁻¹(x)) = 2(x - 1)/2 + 1 = x