Let f(x) = 4x² + x + 1 and g(x) = x² - 2. Find g(f(x)). Show each step of your work.

Solution:

Given: The functions f(x) and g(x)

f(x) = 4x² + x + 1

g(x) = x² - 2 (1)

To find g(f(x)). In the composite function f(x) replaces the x of g(x). 

the output values of f(x) are taken as input for g(x).

g(f(x)) = (4x² + x + 1 )² - 2

Applying (a+b+c)² formula, we have

g(f(x))= [16x⁴ + x² +1 + 2 . 4x². x + 2. x .1 + 2.1.4x²] - 2

=[16x⁴ +  x² + 1 + 8x³ + 2x+ 8x²]- 2

= 16x⁴ + 8x³ + 9x² + 2x + 1 -2

g(f(x)) = 16x⁴ + 8x³ + 9x² + 2x - 1

Let f(x) = 4x² + x + 1 and g(x) = x² - 2. Find g(f(x)). Show each step of your work.

Summary:

The function g(f(x)) = 16x⁴ + 8x³ + 9x² + 2x - 1.