Question 1: Calculate (f o g)(x) where f(x) = 2x + 3 and g(x) = x^2.

Answer: 2x^2 + 3

Question 2: Calculate (g o f)(x) where f(x) = 3x - 1 and g(x) = x^2.

Answer: 9x^2 - 6x + 1

Question 3: If f(x) = x + 4 and g(x) = 5x, find (f o g)(x).

Answer: 5x + 4

Question 4: Evaluate (g o f)(3) where f(x) = 2x and g(x) = x - 1.

Answer: 5

Question 1: A function f(x) = 3x + 2 is applied first, followed by g(x) = x^2. If the result of the composition at x=4 is 144, what is the value of x?

Answer: 4

Question 2: Explain why (f o g)(x) ≠ (g o f)(x) in general, using specific functions f and g as examples.

Answer: Because the order of applying functions changes the intermediate values, affecting the final result.

Question 3: Construct two functions f and g such that (f o g)(x) ≠ (g o f)(x). Show your work.

Answer: f(x)=x+2, g(x)=x^2; (f o g)(x)=(x^2)+2, (g o f)(x)= (x+2)^2

Question 4: A pizza shop combines two processes: adding toppings (f), and baking (g). If adding toppings results in 3 toppings added, and baking doubles the toppings, what is the order effect if toppings are added first then baked versus baked then adding toppings?

Answer: Adding toppings first results in 2*3=6 toppings; baking first results in adding 3 toppings after doubling, which is different.

Question 5: If (f o g)(x) = 4x + 1 and (g o f)(x) = 3x + 5, find functions f and g assuming linear forms.

Answer: f(x)=x+2, g(x)=3x-5

Question 1: A car's journey involves applying function f for acceleration (f(x)=2x) and g for fuel efficiency (g(x)=x/2). If the car first accelerates then checks fuel efficiency, what is the overall effect if starting speed is 10?

Answer: First, speed after acceleration: 2*10=20; then fuel efficiency: 20/2=10

Question 1: Design two non-linear functions f and g such that (f o g)(x) ≠ (g o f)(x) for at least some x. Provide your functions and verify with an example.

Answer: f(x)=x^2+1, g(x)=x+3; verify at x=1

Question 2: Given functions f and g where (f o g)(x)=5x+1 and (g o f)(x)=3x+4, find possible f and g functions, explaining your reasoning.

Answer: f(x)=2x+?, g(x)=? (specific forms depend on additional constraints)

Question 1: True or False: (f o g)(x) = (g o f)(x) for all functions f and g. Justify your answer.

Answer: False

Question 2: Identify the mistake: A student claims that (f o g)(x)=g(f(x)). Is this always true? Explain.

Answer: No, because composition order affects the result; the student confused the order.