To find the value of [tex]\( g(f(x)) \)[/tex], we need to substitute the function [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex].
1. Given functions:
[tex]\[
f(x) = 2x^3 + 3
\][/tex]
[tex]\[
g(x) = x - 4
\][/tex]
2. We need to evaluate [tex]\( g(f(x)) \)[/tex], i.e., to apply function [tex]\( g \)[/tex] to the result of [tex]\( f(x) \)[/tex].
3. First, calculate [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = 2x^3 + 3
\][/tex]
4. Next, substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex] where [tex]\( x \)[/tex] in [tex]\( g(x) \)[/tex] becomes [tex]\( 2x^3 + 3 \)[/tex]:
[tex]\[
g(f(x)) = g(2x^3 + 3)
\][/tex]
Since [tex]\( g(x) = x - 4 \)[/tex], replace [tex]\( x \)[/tex] with [tex]\( 2x^3 + 3 \)[/tex]:
[tex]\[
g(2x^3 + 3) = (2x^3 + 3) - 4
\][/tex]
5. Simplify the expression:
[tex]\[
(2x^3 + 3) - 4 = 2x^3 + 3 - 4 = 2x^3 - 1
\][/tex]
Therefore, the value of [tex]\( g(f(x)) \)[/tex] is:
[tex]\[
g(f(x)) = 2x^3 - 1
\][/tex]
