To find the composition of the functions [tex]\( g \)[/tex] and [tex]\( f \)[/tex], denoted as [tex]\((g \circ f)(x)\)[/tex], we need to substitute the output of the function [tex]\( f \)[/tex] into the function [tex]\( g \)[/tex]. Here's a step-by-step solution:
1. Express [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = 6x - 1
\][/tex]
2. Express [tex]\( g(x) \)[/tex]:
[tex]\[
g(x) = 4x^2 + x
\][/tex]
3. Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex] to find [tex]\( g(f(x)) \)[/tex]:
- First, substitute [tex]\( f(x) \)[/tex] into [tex]\( g \)[/tex]:
[tex]\[
g(f(x)) = g(6x - 1)
\][/tex]
4. Calculate [tex]\( g(6x - 1) \)[/tex]:
- Replace [tex]\( x \)[/tex] in [tex]\( g \)[/tex] with [tex]\( 6x - 1 \)[/tex]:
[tex]\[
g(6x - 1) = 4(6x - 1)^2 + (6x - 1)
\][/tex]
5. Expand and simplify [tex]\( 4(6x - 1)^2 \)[/tex]:
- First, compute [tex]\( (6x - 1)^2 \)[/tex]:
[tex]\[
(6x - 1)^2 = (6x - 1)(6x - 1) = 36x^2 - 12x + 1
\][/tex]
- Then, multiply by 4:
[tex]\[
4(36x^2 - 12x + 1) = 144x^2 - 48x + 4
\][/tex]
6. Combine the results:
[tex]\[
g(6x - 1) = 144x^2 - 48x + 4 + 6x - 1
\][/tex]
- Simplify by combining like terms:
[tex]\[
g(6x - 1) = 144x^2 - 42x + 3
\][/tex]
7. Thus, the composition [tex]\( (g \circ f)(x) \)[/tex] is:
[tex]\[
(g \circ f)(x) = 144x^2 - 42x + 3
\][/tex]
After performing all these steps, you will find that [tex]\((g \circ f)(x) = 144x^2 - 42x + 3\)[/tex]. Evaluating this at [tex]\( x = 1 \)[/tex] can be done to check your work, and the value should confirm the correctness.
