To solve the problem of finding the compositions of functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], we need to follow the given steps:
1. Define the functions:
[tex]\[
f(x) = 4x^2 + 1
\][/tex]
[tex]\[
g(x) = x^2 - 3
\][/tex]
2. Compute [tex]\( (f \circ g)(x) \)[/tex]:
This means we evaluate [tex]\( f \)[/tex] at [tex]\( g(x) \)[/tex].
[tex]\[
(f \circ g)(x) = f(g(x))
\][/tex]
Substitute [tex]\( g(x) \)[/tex] into [tex]\( f \)[/tex]:
[tex]\[
f(g(x)) = f(x^2 - 3)
\][/tex]
Using the definition of [tex]\( f(x) \)[/tex]:
[tex]\[
f(x^2 - 3) = 4(x^2 - 3)^2 + 1
\][/tex]
Simplify inside the parentheses:
[tex]\[
(x^2 - 3)^2 = x^4 - 6x^2 + 9
\][/tex]
Substitute back:
[tex]\[
f(g(x)) = 4(x^4 - 6x^2 + 9) + 1 = 4x^4 - 24x^2 + 36 + 1 = 4x^4 - 24x^2 + 37
\][/tex]
3. Compute [tex]\( (g \circ f)(x) \)[/tex]:
This means we evaluate [tex]\( g \)[/tex] at [tex]\( f(x) \)[/tex].
[tex]\[
(g \circ f)(x) = g(f(x))
\][/tex]
Substitute [tex]\( f(x) \)[/tex] into [tex]\( g \)[/tex]:
[tex]\[
g(f(x)) = g(4x^2 + 1)
\][/tex]
Using the definition of [tex]\( g(x) \)[/tex]:
[tex]\[
g(4x^2 + 1) = (4x^2 + 1)^2 - 3
\][/tex]
Simplify inside the parentheses:
[tex]\[
(4x^2 + 1)^2 = 16x^4 + 8x^2 + 1
\][/tex]
Substitute back:
[tex]\[
g(f(x)) = 16x^4 + 8x^2 + 1 - 3 = 16x^4 + 8x^2 - 2
\][/tex]
So, the compositions are:
[tex]\[
(f \circ g)(x) = 4x^4 - 24x^2 + 37
\][/tex]
[tex]\[
(g \circ f)(x) = 16x^4 + 8x^2 - 2
\][/tex]
Therefore, the correct pairings are:
[tex]\[
(f \circ g)(x) = 4x^4 - 24x^2 + 37
\][/tex]
[tex]\[
(g \circ f)(x) = 16x^4 + 8x^2 - 2
\][/tex]
