Question

Consider the function h(x) = (x - 1)3 + 5(x - 1)2 +6(x - 1) – 4.

Suppose that g(x) = x - 1. Find f(x) such that h(x) = (fºg)(x).

Answer 1

To find f(x) such that h(x) = (fºg)(x), substitute g(x) into h(x) and form the composition, which gives f(x) = h(x) = (x - 1)^3 + 5(x - 1)^2 + 6(x - 1) - 4.

Step-by-step explanation:

To find the function f(x) such that h(x) = (fºg)(x), we need to substitute g(x) = x - 1 into h(x) and form the composition (fºg)(x). Let's start with h(x):

h(x) = (x - 1)^3 + 5(x - 1)^2 + 6(x - 1) - 4

Now, substitute g(x) into h(x):

h(x) = (g(x))^3 + 5(g(x))^2 + 6(g(x)) - 4

Replace g(x) with (x - 1):

h(x) = ((x - 1))^3 + 5((x - 1))^2 + 6((x - 1)) - 4

The resulting function h(x) is equal to (fºg)(x), so f(x) = h(x). Therefore, f(x) = (x - 1)^3 + 5(x - 1)^2 + 6(x - 1) - 4.