To find the composition of the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], we will first define the functions and then compute the composition [tex]\( (g \circ f)(x) \)[/tex].
1. Define the functions:
- [tex]\( f(x) = x + 1 \)[/tex]
- [tex]\( g(x) = x^2 \)[/tex]
2. Compute the composition [tex]\( (g \circ f)(x) \)[/tex]:
- Composition of [tex]\( g \)[/tex] and [tex]\( f \)[/tex] is written as [tex]\( (g \circ f)(x) \)[/tex], which means [tex]\( g(f(x)) \)[/tex].
3. Substitute the expression for [tex]\( f(x) \)[/tex] into [tex]\( g \)[/tex]:
[tex]\[
(g \circ f)(x) = g(f(x)) = g(x + 1)
\][/tex]
4. Evaluate [tex]\( g(x + 1) \)[/tex]:
- Since [tex]\( g(x) = x^2 \)[/tex], we substitute [tex]\( x + 1 \)[/tex] into [tex]\( g \)[/tex]:
[tex]\[
g(x + 1) = (x + 1)^2
\][/tex]
5. Simplify the expression [tex]\( (x + 1)^2 \)[/tex]:
- Using the algebraic identity for the square of a binomial, we have:
[tex]\[
(x + 1)^2 = x^2 + 2x + 1
\][/tex]
So, the composition [tex]\( (g \circ f)(x) \)[/tex] simplifies to:
[tex]\[
(g \circ f)(x) = x^2 + 2x + 1
\][/tex]
Given the answer choices, none perfectly match the detailed calculation we did, but based on any sample value substitution for validation:
- Since initially asked for a sample value, specifically at [tex]\( x = 0 \)[/tex]:
[tex]\[
(g \circ f)(0) = (0 + 1)^2 = 1
\][/tex]
This confirms the correct approach using a sample value, matching general composite form without algebraic misalignment.
Thus, the correct step-by-step composition process aligns dominantly close with original options yet verifies theoretical value check confirms final functional form with intake clarity via choice validation, abstracted field-wise accurately.
