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

Answer: 16

Question 2: Calculate (g∘f)(2) with the same functions as above.

Answer: 20

Question 3: Explain why (f∘g)(x) ≠ (g∘f)(x) in general, using the given functions.

Answer: Because composition depends on the order; f(g(x)) and g(f(x)) apply functions in different sequences, leading to different results.

Question 4: Construct a pair of functions f and g such that (f∘g)(x) = 2x + 5 and (g∘f)(x) ≠ 2x + 5.

Answer: An example is f(x)=x+3, g(x)=2x+2. Then, (f∘g)(x)=f(2x+2)=2x+5, but (g∘f)(x)=g(x+3)=2(x+3)+2=2x+8.

Question 5: A function h(x) = 5−x. If f(x)=2x and g(x)=x+1, find the value of (h∘f)(3).

Answer: 3

Question 6: Calculate (g∘h)(4) where h(x) = 3x−2 and g(x) = x^2 + 1.

Answer: 97

Question 7: Identify the common mistake students make when reversing the order of functions in composition and explain how to avoid it.

Answer: A common mistake is to assume (f∘g)(x) = (g∘f)(x). To avoid this, students should practice evaluating both compositions step-by-step and understand that order affects the result.

Question 8: Given f(x)=x−2 and g(x)=4x, find (f∘g)(x) and (g∘f)(x). Show that they are different.

Answer: f(g(x))=4x−2, g(f(x))=4(x−2)=4x−8

Question 9: Plot the graph of y = (f∘g)(x) where f(x)=x+1 and g(x)=2x on the provided grid.

Answer: The graph of y=2x+1.

Question 10: Construct two functions such that (f∘g)(x)=3x and (g∘f)(x)=x+4. Explain why this is possible.

Answer: Because the functions are designed with specific compositions that produce different results depending on order, illustrating non-commutativity.

Question 11: If students mistakenly compute (f∘g)(x) as g(f(x)), what misconception are they demonstrating?

Answer: They are confusing composition order, assuming the operation is commutative, which it is not.

Question 12: Challenge: Find functions p and q such that (p∘q)(x)=x^3 + 2 and (q∘p)(x)=x^2 + 5. Show your reasoning.

Answer: One approach: p(x)=x^3+2, q(x)=x^2+3. Then, (p∘q)(x)=(x^2+3)^3+2, and (q∘p)(x)=(x^3+2)^2+5. Adjust functions accordingly to meet the specified outputs.