Use the definition of inverses to determine whether and are inverses.

Are fand inverses of each other?
Yes

Ask by Barrett Hanson. in the United States

Nov 04,2024

Question

Use the definition of inverses to determine whether and are inverses.

Are fand inverses of each other?
Yes

Ask by Barrett Hanson. in the United States

Nov 04,2024

UpStudy AI · Accepted Answer

To determine if two functions are inverses of each other, we need to check if the composition of the two functions results in the identity function, which is $$ f(g(x)) = x $$ and $$ g(f(x)) = x $$.

Given:
$$ f(x) = x^{2} + 2 $$ for $$ x \geq 0 $$
$$ g(x) = \sqrt{x-2} $$ for $$ x \geq 2 $$

Let's first find $$ f(g(x)) $$:
$$ f(g(x)) = f(\sqrt{x-2}) = (\sqrt{x-2})^{2} + 2 = x - 2 + 2 = x $$

Next, let's find $$ g(f(x)) $$:
$$ g(f(x)) = g(x^{2} + 2) = \sqrt{(x^{2} + 2) - 2} = \sqrt{x^{2}} = x $$

Since both $$ f(g(x)) = x $$ and $$ g(f(x)) = x $$, we can conclude that $$ f $$ and $$ g $$ are inverses of each other.
$$ f(g(x)) = x $$ and $$ g(f(x)) = x $$, so $$ f $$ and $$ g $$ are inverses.

Use the definition of inverses to determine whether and are inverses.

Are fand inverses of each other?
Yes

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Use the definition of inverses to determine whether and are inverses. Are fand inverses of each other? Yes

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Use the definition of inverses to determine whether and are inverses.

Are fand inverses of each other?
Yes