Question 1: Calculate the value of (f ◦ g)(2) if f(x) = 3x + 4 and g(x) = x^2.

Answer: 20

Question 2: Evaluate (g ◦ f)(-1) where f(x) = 2x - 5 and g(x) = x + 7.

Answer: 4

Question 3: Plot the graph of y = 2x on the grid.

Answer: Student draws the line y=2x

Question 4: Construct a triangle with vertices at (1,2), (4,2), and (3,5).

Answer: Student draws the triangle on grid

Question 5: If f(x) = x + 5 and g(x) = 2x, what is (f ◦ g)(3)?

Answer: 11

Question 6: Explain why (f ◦ g)(x) may not equal (g ◦ f)(x) with an example.

Answer: Because the order of composition changes the order of applying functions, leading to different results, e.g., with f(x)=x+2 and g(x)=3x, (f ◦ g)(x)=f(3x)=3x+2, while (g ◦ f)(x)=g(x+2)=3(x+2)=3x+6.

Question 7: A recipe calls for mixing ingredients in a specific order: first add ingredient A, then ingredient B. If you reverse the order, the final dish changes. What does this illustrate about composite functions?

Answer: It illustrates that order matters in composition; changing the order of operations affects the result.

Question 8: Find (f ◦ g)(x) where f(x) = 4x - 1 and g(x) = x/2. Then, find (g ◦ f)(x). Are these functions the same?

Answer: f◦g(x)=2x-1; g◦f(x)=2x-0.5; No, they are not the same.

Question 9: A student claims that (f ◦ g)(x) always equals (g ◦ f)(x). Give an example where this is false.

Answer: Using f(x)=x+1 and g(x)=2x, (f ◦ g)(x)=2x+1, while (g ◦ f)(x)=2(x+1)=2x+2, which are not equal.

Question 10: Identify the common mistake made when students forget to switch the order of functions when calculating composite functions.

Answer: They mistakenly treat composition as commutative, calculating (f ◦ g)(x) as (g ◦ f)(x).

Question 11: Calculate (f ◦ g)(x) if f(x)=x^2 + 1 and g(x)=x - 3.

Answer: x^2 - 6x + 10