Sure! Let's determine the composite function [tex]\( g \cdot f \)[/tex], which is represented as [tex]\( g(f(x)) \)[/tex].
Given:
[tex]\[ f(x) = x^2 - 3 \][/tex]
[tex]\[ g(x) = x + 1 \][/tex]
First, we need to find [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = x^2 - 3 \][/tex]
Next, we apply the function [tex]\( g \)[/tex] to the result of [tex]\( f(x) \)[/tex]:
[tex]\[ g(f(x)) = g(x^2 - 3) \][/tex]
Since the function [tex]\( g \)[/tex] is defined as [tex]\( g(x) = x + 1 \)[/tex], we substitute [tex]\( x^2 - 3 \)[/tex] into [tex]\( g \)[/tex]:
[tex]\[ g(x^2 - 3) = (x^2 - 3) + 1 \][/tex]
Simplifying the expression, we get:
[tex]\[ g(x^2 - 3) = x^2 - 3 + 1 \][/tex]
[tex]\[ g(x^2 - 3) = x^2 - 2 \][/tex]
Hence, the composite function [tex]\( g \cdot f \)[/tex] is:
[tex]\[ g(f(x)) = x^2 - 2 \][/tex]
Let's verify this composite function with an example input, [tex]\( x = 2 \)[/tex]:
First, calculate [tex]\( f(2) \)[/tex]:
[tex]\[ f(2) = 2^2 - 3 \][/tex]
[tex]\[ f(2) = 4 - 3 \][/tex]
[tex]\[ f(2) = 1 \][/tex]
Next, apply the result to [tex]\( g \)[/tex]:
[tex]\[ g(f(2)) = g(1) \][/tex]
[tex]\[ g(1) = 1 + 1 \][/tex]
[tex]\[ g(1) = 2 \][/tex]
Thus, for [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = 1 \][/tex]
[tex]\[ g(f(2)) = 2 \][/tex]
So, the result is:
[tex]\[ (1, 2) \][/tex]
Therefore, the composite function [tex]\( g \cdot f \)[/tex] is correctly [tex]\( x^2 - 2 \)[/tex].
