Question 1: Recall the general form of a composite function f(g(x)). Write the expression for f(g(x)) if f(x) = 2x + 3 and g(x) = x^2 - 1.

Answer: f(g(x)) = 2(x^2 - 1) + 3

Question 2: Explain in your own words what the composite function f(g(x)) represents.

Answer: It represents applying g(x) first, then applying f to the result.

Question 1: Calculate f(g(2)) for f(x) = 3x - 4 and g(x) = x + 5.

Answer: f(g(2))= 3(2+5)-4= 3(7)-4= 21-4=17

Question 2: If f(x) = x^2 + 2 and g(x) = 4 - x, find f(g(3)).

Answer: f(g(3))= (4-3)^2 + 2= 1^2 + 2= 1 + 2= 3

Question 3: Given f(x) = 2x + 1 and g(x) = x/2, what is f(g(8))?

Answer: f(g(8))= 2(8/2)+1= 2(4)+1= 8+1= 9

Question 1: The functions are f(x) = 5x - 2 and g(x) = x^2 + 1. Find a simplified expression for f(g(x)).

Answer: f(g(x))= 5(x^2 + 1) - 2= 5x^2 + 5 - 2= 5x^2 + 3

Question 2: If the composite function f(g(x)) has a constant value for all x, what does this imply about the functions f and g?

Answer: It implies that g(x) is a constant value, leading to f being constant or g mapping all x to a value that f maps to a constant.

Question 3: Given f(x) = 4x + 7 and g(x) = 3x - 5, find f(g(4)). Show your working.

Answer: g(4)= 3(4)-5=12-5=7; f(7)=4(7)+7=28+7=35

Question 1: A car rental company charges a fixed fee plus a rate per mile. The total cost C(x) = 50 + 0.2x. If g(x) models the miles driven as x, and f(x) models the total cost after applying a discount function h(x)=x-10 on miles driven, write an expression for the total cost after the discount, expressed as a composite function.

Answer: Total cost = f(h(g(x)))= 50 + 0.2(g(x)-10)= 50 + 0.2(x-10)= 50 + 0.2x - 2

Question 1: Given functions f(x) = x^3 and g(x) = 2x + 1, find the expression for f(g(g(2))).

Answer: g(2)= 2(2)+1=5; g(5)= 2(5)+1=11; f(11)= 11^3=1331

Question 2: Identify and correct the mistake in the following student solution: f(x) = x+4, g(x) = x^2, compute f(g(3)) and the student wrote: f(g(3))= (3)^2+4=9+4=13, but claims the answer is 16.

Answer: Correct calculation: f(g(3))= 9+4=13. Mistake: the student incorrectly claimed the answer is 16; the correct answer is 13.

Question 1: Construct a function g(x) and a function f(x) such that f(g(x)) = 3x + 2, but neither f nor g is linear.

Answer: One possibility: g(x)=x^2, f(x)=3√x + 2 for x≥0, so that f(g(x))=3x + 2.

Question 2: Plot the graph of y=2x on the provided grid. Then, describe how the graph of y=f(g(x)) might look if f and g are both linear functions with slopes of 2 and 3 respectively.

Answer: Graph y=2x is a straight line through origin with slope 2. If g(x)=3x, f(x)=2x, then f(g(x))= 2(3x)=6x, a line with slope 6.