Question 1: If f(x) = 3x + 4, find the inverse function f⁻¹(x).

Answer: f⁻¹(x) = (x - 4) / 3

Question 2: Calculate f⁻¹(10) for the function f(x) = 2x - 6.

Answer: 8

Question 3: Given the inverse function f⁻¹(x) = (x + 5) / 2, find the original function f(x).

Answer: f(x) = 2x - 5

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

Answer: Student draws y = x/2 reflected across y=x

Question 5: Construct a triangle on the grid with sides representing a sound wave where frequency is inversely proportional to wavelength. Use f(x) = 1000 / x to model this, with x representing wavelength.

Answer: Student constructs and demonstrates inverse relation

Question 6: A sound engineer uses the function f(x) = 12000 / x to determine the frequency (Hz) for a given wavelength (nm). Find the inverse function to determine wavelength from frequency.

Answer: f⁻¹(x) = 12000 / x

Question 7: A music app maps volume (x) to perceived loudness (f(x)) with f(x) = 10x. Find the inverse function to determine the volume from loudness.

Answer: f⁻¹(x) = x / 10

Question 8: A sound wave's frequency f (Hz) relates to wavelength λ (meters) by f(λ) = 340 / λ. Write the inverse function to find wavelength from frequency.

Answer: f⁻¹(x) = 340 / x

Question 9: Identify and correct the mistake in this inverse function: f⁻¹(x) = (x + 4) / 2, given that the original function is f(x) = 2x - 4.

Answer: The inverse should be f⁻¹(x) = (x + 4) / 2; no mistake

Question 10: A sound technician measures the frequency of a sound wave with the function f(x) = 5000 / x, where x is the wavelength. If the measured frequency is 1000 Hz, what is the wavelength? Show your calculation.

Answer: x = 5 meters