Question

Drag the expressions to the correct functions. Not all expressions will be used.
Consider the functions fand g.
= 4x² + 1
g(x) =
Perform the function compositions:
x² – 3

1. The function composition exists an operation ” ∘ ” that brings two functions f and g, and has a function h = g ∘ f such that h(x) = g(f(x)).
Let the functions be f(x) = 4x² + 1 and g(x) = x² – 3
The correct answer is (f o g)(x) = 4x⁴ – 96x + 37 and
(g o f)(x) = 16x⁴ + 8x² – 2.

### What is composition function?

The function composition exists an operation ” ∘ ” that brings two functions f and g, and has a function h = g ∘ f such that h(x) = g(f(x)). In this operation, the function g exists used for the outcome of applying the function f to x.
Given:
f(x) = 4x² + 1 and g(x) = x² – 3
a) (f o g)(x) = f[g(x)]
f[g(x)] = 4(x² – 3)² + 1
substitute the value of g(x) in the above equation, and we get
= 4(x⁴ – 24x + 9) + 1
simplifying the above equation
= 4x⁴ – 96x + 36 + 1
= 4x⁴ – 96x + 37
(f o g)(x) = 4x⁴ – 96x + 37
b) (g o f)(x) = g[f(x)]
substitute the value of g(x) in the above equation, and we get
g[f(x)] = (4x² + 1)²- 3
= 16x⁴ + 8x² + 1 – 3
simplifying the above equation
= 16x⁴ + 8x² – 2
(g o f)(x) = 16x⁴ + 8x² – 2.
Therefore, the correct answer is (f o g)(x) = 4x⁴ – 96x + 37 and
(g o f)(x) = 16x⁴ + 8x² – 2.