Question 1: The inverse function f⁻¹(x) reverses the process of f(x) such that if y = f(x), then x = f⁻¹(y). To find f⁻¹(x), solve the equation y = f(x) for x in terms of y, then replace y with x. Remember, the inverse function undoes what the original function does.

Answer: f⁻¹(x) is found by solving y = f(x) for x, then swapping x and y.

Question 1: Given the function f(x) = 3x + 5, find the inverse function f⁻¹(x).

Answer: f⁻¹(x) = (x - 5) / 3

Question 2: For the function f(x) = (2x - 4)/3, verify that its inverse f⁻¹(x) = (3x + 4)/2 is correct by composing f and f⁻¹ and showing the result is x.

Answer: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x

Question 3: If f(x) = x² + 1, find the inverse function, considering the domain restriction x ≥ 0.

Answer: f⁻¹(x) = √(x - 1), with x ≥ 1

Question 4: Plot the graph of y = 2x and then draw the graph of its inverse on the same grid.

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

Question 5: Construct a triangle with sides 3cm, 4cm, and 5cm on the grid. Describe how the inverse function could help in transforming the lengths if the scale is changed.

Answer: Inverse functions can be used to find original lengths after scale transformation

Question 6: Solve for x: 5f(x) + 2 = 17, where f(x) = 2x - 3. Find x using inverse functions.

Answer: x = 4

Question 7: A function f(x) = (x + 2)/3 is given. Find its inverse and verify it by substitution.

Answer: f⁻¹(x) = 3x - 2, verified by substitution

Question 8: A real-world scenario: A savings account increases by 4% annually. If the amount after x years is given by f(x) = 1000(1.04)^x, find the inverse function to determine the number of years from the final amount.

Answer: x = log(final amount / 1000) / log(1.04)

Question 9: A function f(x) = 2x + 7 is transformed into an inverse function. Calculate and graph the inverse function.

Answer: f⁻¹(x) = (x - 7)/2

Question 10: Identify and correct the mistake in this inverse function attempt: f(x) = 5x - 2, inverse claimed as f⁻¹(x) = (x + 2)/5.

Answer: The correct inverse is f⁻¹(x) = (x + 2)/5; verify by composition