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Using the composition of functions, determine if the two functions are inverses of each other.

[tex]\[
\begin{array}{l}
F(x) = \sqrt{x} - 6 \\
G(x) = (x + 6)^2
\end{array}
\][/tex]

A. No, because the functions contain different operations.
B. No, because the composition does not result in an answer of [tex]$x$[/tex].
C. Yes, because [tex]$F(x)$[/tex] is equal to [tex]$-G(x)$[/tex].
D. Yes, but only within the domain [tex]$x \geq 0$[/tex].


Sagot :

To determine if two functions are inverses of each other, we need to examine their composition in both possible orders: [tex]\( F(G(x)) \)[/tex] and [tex]\( G(F(x)) \)[/tex]. If both compositions simplify to [tex]\( x \)[/tex], then the functions are indeed inverses of each other.

Let's denote the given functions:
[tex]\[ F(x) = \sqrt{x} - 6 \][/tex]
[tex]\[ G(x) = (x + 6)^2 \][/tex]

### Finding [tex]\( F(G(x)) \)[/tex]:

First, we'll substitute [tex]\( G(x) \)[/tex] into [tex]\( F(x) \)[/tex]:
[tex]\[ F(G(x)) = F((x + 6)^2) \][/tex]
Substituting [tex]\( (x + 6)^2 \)[/tex] into [tex]\( F(x) \)[/tex]:
[tex]\[ F((x + 6)^2) = \sqrt{(x + 6)^2} - 6 \][/tex]

Now simplify [tex]\( \sqrt{(x + 6)^2} \)[/tex]:
[tex]\[ \sqrt{(x + 6)^2} = |x + 6| \][/tex]

Thus,
[tex]\[ F(G(x)) = |x + 6| - 6 \][/tex]

Given [tex]\( G(x) = (x + 6)^2 \)[/tex] which is always non-negative for real [tex]\( x \)[/tex], and considering the square root operation returns the principal (non-negative) square root,
[tex]\[ |x + 6| = x + 6 \][/tex]
So,
[tex]\[ F(G(x)) = x + 6 - 6 = x \][/tex]

Hence,
[tex]\[ F(G(x)) = x \][/tex]

### Finding [tex]\( G(F(x)) \)[/tex]:

Next, substitute [tex]\( F(x) \)[/tex] into [tex]\( G(x) \)[/tex]:
[tex]\[ G(F(x)) = G(\sqrt{x} - 6) \][/tex]
Substituting [tex]\( \sqrt{x} - 6 \)[/tex] into [tex]\( G(x) \)[/tex]:
[tex]\[ G(\sqrt{x} - 6) = (\sqrt{x} - 6 + 6)^2 = (\sqrt{x})^2 \][/tex]

Simplifying,
[tex]\[ (\sqrt{x})^2 = x \][/tex]

Thus,
[tex]\[ G(F(x)) = x \][/tex]

### Conclusion:

Since both compositions [tex]\( F(G(x)) \)[/tex] and [tex]\( G(F(x)) \)[/tex] simplify to [tex]\( x \)[/tex], we conclude that [tex]\( F(x) \)[/tex] and [tex]\( G(x) \)[/tex] are indeed inverses of each other.

Therefore, the appropriate answer is:

D. Yes, but only within the domain [tex]\( x \geq 0 \)[/tex].

The domain constraint is due to the square root function [tex]\( \sqrt{x} \)[/tex], which requires [tex]\( x \geq 0 \)[/tex] to be defined in the real numbers.
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