To determine [tex]$(g \circ f)(x)$[/tex], we need to find the composition of the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].
Given:
[tex]\[ f(x) = x + 1 \][/tex]
[tex]\[ g(x) = x^2 \][/tex]
The notation [tex]\( (g \circ f)(x) \)[/tex] means we apply [tex]\( f \)[/tex] first and then [tex]\( g \)[/tex] to the result of [tex]\( f \)[/tex]. In other words:
[tex]\[ (g \circ f)(x) = g(f(x)) \][/tex]
Step-by-step solution:
1. Start by applying [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = x + 1 \][/tex]
2. Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(f(x)) = g(x + 1) \][/tex]
3. Now substitute [tex]\( x + 1 \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(x + 1) = (x + 1)^2 \][/tex]
Therefore, the composition [tex]\( (g \circ f)(x) \)[/tex] is:
[tex]\[ (g \circ f)(x) = (x + 1)^2 \][/tex]
Among the given options, the correct one is:
[tex]\[ (g \circ f)(x) = (x + 1)^2 \][/tex]
