Function Composition Exploration

Function Composition Exploration

Read each question carefully and show all your work. Simplify your answers.

1. Function composition is the application of one function to the results of another, denoted as (f ∘ g)(x) or f(g(x)). This means the   of the inner function becomes the   of the outer function.

2. When composing functions, the   of the inner function must overlap with the   of the outer function for the composition to be defined.

Given f(x) = 2x + 3 and g(x) = x² - 1, find:

3. (f ∘ g)(x)

4. (g ∘ f)(x)

5. (f ∘ g)(2)

6. If h(x) = √(x) and k(x) = x + 5, what is (h ∘ k)(x)?

7. A store offers a 20% discount on all items. On Tuesdays, they offer an additional $5 off. Let d(x) = x - 0.20x be the discount function and t(x) = x - 5 be the Tuesday discount function.

a) If you apply the 20% discount first, then the $5 Tuesday discount, what is the combined function (t ∘ d)(x)?

b) If you apply the $5 Tuesday discount first, then the 20% discount, what is the combined function (d ∘ t)(x)?

c) Which order of discounts gives a better deal for an item that costs $50? Explain your reasoning.