Question 1: Calculate (f∘g)(3) where f(x) = 2x + 5 and g(x) = x^2.

Answer: 23

Question 2: If f(x) = 3x - 4 and g(x) = x + 7, what is (g∘f)(2)?

Answer: 13

Question 3: Construct a composition where (f∘g)(x) results in 5x + 2. Given g(x) = 2x, find f(x).

Answer: f(x) = (5x + 2)/2

Question 4: Plot the graph of y = (g∘f)(x), where f(x) = x + 3 and g(x) = 2x.

Answer: y = 2(x + 3) = 2x + 6

Question 1: Find (f∘g)(4) where f(x) = x^2 + 1 and g(x) = 3x.

Answer: 37

Question 2: Calculate (g∘f)(-2) with f(x) = 2x - 1 and g(x) = x/2.

Answer: -2

Question 3: Given f(x) = x - 4 and g(x) = 3x, find (f∘g)(2).

Answer: 2

Question 1: A bakery uses two functions: f(x) = x + 10 for the number of cupcakes added daily, and g(x) = 2x for the total cupcakes after baking. If the bakery starts with 20 cupcakes, what is the total after applying g∘f? Explain the steps.

Answer: First apply f: 20 + 10 = 30. Then g: 2 × 30 = 60. Total cupcakes = 60.

Question 1: A car's speed is modeled by f(x) = 60 + 5x (where x is hours), and the distance traveled is g(x) = speed × time. Find the total distance after 4 hours if you first calculate the speed change in f and then find the distance in g. Explain the process.

Answer: Speed after 4 hours: 60 + 5×4 = 80 mph. Distance: 80 × 4 = 320 miles.

Question 1: Given f(x) = 3x + 2 and g(x) = x^2, find (g∘f)(x). Determine the result when x=2. Why does the order change the answer?

Answer: g(f(2)) = (3×2 + 2)^2 = (6 + 2)^2 = 8^2 = 64. Order matters because (f∘g)(2) = f(g(2)) = f(4) = 14.

Question 2: Create a composition such that (f∘g)(x) = 4x + 1, with g(x) = 2x. Find f(x).

Answer: f(x) = (4x + 1)/2

Question 1: Identify and correct the mistake: If f(x) = x + 3 and g(x) = 2x, students compute (g∘f)(x) as 2(x + 3) = 2x + 3.

Answer: Incorrect; 2(x + 3) = 2x + 6, not 2x + 3.

Question 2: Plot the graph of (g∘f)(x) where f(x) = x - 1 and g(x) = 3x.

Answer: y = 3(x - 1) = 3x - 3