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Using the following equations, find [tex]$g(f(x))$[/tex]:

[tex]\[
\begin{array}{r}
f(x) = x^2 - 1 \\
g(x) = x + 2 \\
g(f(x)) = [?]
\end{array}
\][/tex]

Sagot :

GinnyAnsweridn
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]
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