Certainly! Let's proceed step-by-step to find the formula for the composition of the functions [tex]\( (g \circ f)(x) \)[/tex] and then simplify the expression.
1. Define the functions:
We have:
[tex]\[
f(x) = x + 5
\][/tex]
[tex]\[
g(x) = x^2 - 2
\][/tex]
2. Form the composition [tex]\( (g \circ f)(x) \)[/tex]:
The composition [tex]\( (g \circ f)(x) \)[/tex] means [tex]\( g(f(x)) \)[/tex]. First, we need to apply [tex]\( f(x) \)[/tex], and then take the result and apply [tex]\( g \)[/tex] to it.
So, we first compute [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = x + 5
\][/tex]
3. Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
Next, we need to substitute [tex]\( f(x) = x + 5 \)[/tex] into the function [tex]\( g(x) \)[/tex]:
[tex]\[
g(f(x)) = g(x + 5)
\][/tex]
Now, we substitute [tex]\( x + 5 \)[/tex] into [tex]\( g(x) = x^2 - 2 \)[/tex]:
[tex]\[
g(x + 5) = (x + 5)^2 - 2
\][/tex]
4. Simplify the expression [tex]\( (x + 5)^2 \)[/tex]:
To find [tex]\( (x + 5)^2 \)[/tex], expand it as follows:
[tex]\[
(x + 5)^2 = x^2 + 10x + 25
\][/tex]
5. Combine and simplify further:
Substitute [tex]\( (x + 5)^2 \)[/tex] back into the expression for [tex]\( g(x + 5) \)[/tex]:
[tex]\[
g(x + 5) = x^2 + 10x + 25 - 2
\][/tex]
Simplify by combining the constant terms:
[tex]\[
g(x + 5) = x^2 + 10x + 23
\][/tex]
Thus, the formula for [tex]\( (g \circ f)(x) \)[/tex] is:
[tex]\[
(g \circ f)(x) = x^2 + 10x + 23
\][/tex]
This is the simplified expression for the composition of the given functions.
