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

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

Question 2: Verify whether the function g(x) = (2x - 4)/5 has an inverse. If yes, find it.

Answer: Yes, inverse is g^{-1}(x)=(5x+4)/2

Question 3: Construct the inverse of h(x) = x^2 for x ≥ 0.

Answer: h^{-1}(x)=√x

Question 4: If f(x) = (x+2)/(3x-4), find its inverse.

Answer: f^{-1}(x)=(4x+2)/(1-3x)

Question 1: A car rental company charges a flat fee plus a rate per mile. The total cost C(x) in dollars for x miles is given by C(x) = 50 + 0.75x. Find the inverse function to determine how many miles can be driven for a given total cost.

Answer: C^{-1}(x)=(x-50)/0.75

Question 2: A gym tracks the number of visits (v) based on the membership fee (f) using the function f(v)=20v + 30. Verify whether this function has an inverse and find it if it exists.

Answer: Yes, inverse is f^{-1}(x)=(x-30)/20

Question 3: A water tank's volume V is related to the height h of the water level by V=πr^2h, assuming r is constant. If the volume is known, explain how to find the height of the water.

Answer: Solve for h: h=V/(πr^2)

Question 4: The temperature T in °C is related to the Fahrenheit F by T=(F-32)*5/9. Find the inverse function to convert Celsius back to Fahrenheit.

Answer: F= (9/5)T + 32

Question 1: A mobile phone plan charges a fixed monthly fee plus a cost per minute. The total monthly bill B(x) in dollars for x minutes is B(x)=25 + 0.10x. How many minutes can a customer use if their bill is $45?

Answer: x= (45-25)/0.10=200

Question 2: A real estate agent calculates property value V based on the number of bedrooms n using V(n) = 50,000 + 10,000n. Find the inverse function to determine the number of bedrooms based on property value.

Answer: n= (V - 50,000)/10,000

Question 3: A delivery service charges based on weight w in kg, with a cost function C(w) = 15 + 2w. If a customer pays $31, how much weight was shipped?

Answer: w= (31-15)/2=8 kg

Question 1: The function f(x) = (3x+1)/(x-2) is given. Show algebraically whether it has an inverse, and if so, find it.

Answer: Yes, inverse is f^{-1}(x)=(2x+1)/(x+3)

Question 2: Verify if the function g(x)=x^3 + 4x is invertible. If yes, find g^{-1}(x).

Answer: Yes, inverse found via solving y=x^3+4x

Question 1: A function f(x)=2x+3 is claimed to have an inverse f^{-1}(x)= (x-3)/2. Is this correct? If not, identify and correct the mistake.

Answer: Yes, the inverse is correct.

Question 2: A student finds the inverse of h(x)=x^2 and states h^{-1}(x)=x^2. Is this correct? Explain.

Answer: No, because h(x)=x^2 is not invertible over all real numbers; restricted domain needed.