To find the composition of the functions [tex]\( g(f(x)) \)[/tex] using the given functions:
1. Write down the given functions:
- [tex]\( f(x) = x^2 - 1 \)[/tex]
- [tex]\( g(x) = x + 2 \)[/tex]
2. Substitute [tex]\( f(x) \)[/tex] into [tex]\( g \)[/tex]:
The composition [tex]\( g(f(x)) \)[/tex] means that we need to substitute [tex]\( f(x) \)[/tex] into [tex]\( g \)[/tex].
3. Compute [tex]\( g(f(x)) \)[/tex]:
To find [tex]\( g(f(x)) \)[/tex], we first find [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = x^2 - 1
\][/tex]
Now we substitute [tex]\( f(x) \)[/tex] into [tex]\( g \)[/tex]:
[tex]\[
g(f(x)) = g(x^2 - 1)
\][/tex]
4. Substitute [tex]\( x^2 - 1 \)[/tex] into [tex]\( g(x) \)[/tex]:
The function [tex]\( g(x) = x + 2 \)[/tex] needs [tex]\( x \)[/tex] replaced with [tex]\( x^2 - 1 \)[/tex]:
[tex]\[
g(x^2 - 1) = (x^2 - 1) + 2
\][/tex]
5. Simplify the expression:
Simplify the expression [tex]\( (x^2 - 1) + 2 \)[/tex]:
[tex]\[
(x^2 - 1) + 2 = x^2 - 1 + 2 = x^2 + 1
\][/tex]
Thus, the composition [tex]\( g(f(x)) \)[/tex] is [tex]\( x^2 + 1 \)[/tex].
Therefore:
[tex]\[
g(f(x)) = x^2 + 1
\][/tex]
