To solve for the composite function [tex]\((g \circ f)(x)\)[/tex], we need to apply the function [tex]\(f(x)\)[/tex] first and then apply the function [tex]\(g(x)\)[/tex] to the result of [tex]\(f(x)\)[/tex].
1. Define [tex]\( f(x) \)[/tex]:
[tex]\( f(x) = 6x - 4 \)[/tex]
2. Define [tex]\( g(x) \)[/tex]:
[tex]\( g(x) = 2x^2 + 5 \)[/tex]
3. Find [tex]\( f(x) \)[/tex]:
[tex]\( f(x) = 6x - 4 \)[/tex]
4. Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\( g(f(x)) = g(6x - 4) \)[/tex]
5. Evaluate [tex]\( g(6x - 4) \)[/tex]:
Substitute [tex]\( 6x - 4 \)[/tex] into [tex]\( g \)[/tex]:
[tex]\( g(6x - 4) = 2(6x - 4)^2 + 5 \)[/tex]
6. Expand [tex]\( (6x - 4)^2 \)[/tex]:
[tex]\((6x - 4)^2 = 36x^2 - 48x + 16\)[/tex]
7. Multiply the result by 2:
[tex]\( 2 \cdot (36x^2 - 48x + 16) = 72x^2 - 96x + 32 \)[/tex]
8. Add 5 to the result:
[tex]\( 72x^2 - 96x + 32 + 5 = 72x^2 - 96x + 37 \)[/tex]
Therefore, the composite function is:
[tex]\[ (g \circ f)(x) = 72x^2 - 96x + 37 \][/tex]
