Question 1: Given f(x) = 2x + 3 and g(x) = x^2, find (f ◦ g)(4).

Answer: 49

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

Answer: 7

Question 3: Evaluate (f ◦ g)(x) for f(x) = x/2 + 1 and g(x) = 4x, then find the value when x=5.

Answer: 11

Question 4: Construct a composite function where f(x) = 3x and g(x) = x + 5. Calculate (g ◦ f)(2).

Answer: 13

Question 5: Explain why (f ◦ g)(x) may give a different result from (g ◦ f)(x) in general, using functions f(x)=x+2 and g(x)=2x.

Answer: Because composition depends on the order: (f ◦ g)(x)=f(g(x))=2x+2, while (g ◦ f)(x)=g(x+2)=2(x+2)=2x+4, which are different expressions.

Question 6: Plot the graph of y=f(g(x)) where f(x)=x^2 and g(x)=x+1 for x in [-3,3].

Answer: Students plot (x+1)^2 on the grid.

Question 7: Construct a triangle with vertices at (0,0), (2,4), and (4,1) by drawing on the grid. Identify the order of points to form a proper triangle.

Answer: Order: (0,0) → (2,4) → (4,1) → (0,0).

Question 8: Given f(x)=x+3 and g(x)=x/2, verify whether (f ◦ g)(x) equals (g ◦ f)(x).

Answer: No, (f ◦ g)(x)=f(g(x))=x/2+3, while (g ◦ f)(x)=g(x+3)=(x+3)/2.

Question 9: A real-world scenario: A factory produces items where process A triples the item count, then process B adds 10. If starting with 5 items, how many are there after both processes in order A then B? What if the order is B then A?

Answer: Order A then B: (5×3)+10=25. Order B then A: (5+10)×3=45.

Question 10: Identify and correct the mistake in this statement: "Since f(g(x))=g(f(x)) in all cases, order does not matter in composition."

Answer: The mistake is that composition is generally not commutative; f(g(x)) ≠ g(f(x)) unless specific functions are chosen.