Question

Given that f(x)=x2+9x and g(x)=x−1, calculate

(a) f∘g(x)=
(b) g∘f(x)=
(c) f∘f(x)=
(d) g∘g(x)=

Answer 1

To find the composition of functions, you substitute one function into the other as required. f∘g(x) results in x² + 7x - 8, g∘f(x) gives x² + 9x - 1, f∘f(x) yields x⁴ + 18x³ + 90x² + 81x, and g∘g(x) simplifies to x - 2.

Step-by-step explanation:

The question involves the composition of functions, specifically finding the result when one function is applied after another. This is a key concept in algebra and precalculus.

To find f∘g(x), we substitute g(x) into f(x):

f(g(x)) = f(x-1) = (x-1)² + 9(x-1) = x² - 2x + 1 + 9x - 9 = x² + 7x - 8

To find g∘f(x), we substitute f(x) into g(x):

g(f(x)) = g(x² + 9x) = x² + 9x - 1

To find f∘f(x), we substitute f(x) into itself:

f(f(x)) = f(x² + 9x) = (x² + 9x)² + 9(x² + 9x) = x⁴ + 18x³ + 81x² + 9x² + 81x

To find g∘g(x), we substitute g(x) into itself:

g(g(x)) = g(x - 1) = (x - 1) - 1 = x - 2