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Consider the following functions:

[tex]\[ f(x) = x + 5 \][/tex]
[tex]\[ g(x) = x^2 - 2 \][/tex]

Find the formula for [tex]\((g \circ f)(x)\)[/tex] and simplify your answer.


Sagot :

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.
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