How do you find the compositions given $f (x) = \sqrt{x - 3}$ $f(x)=sqrt(x-3)$ , $g (x) = \sqrt{x + 3}$ $g(x)=sqrt(x +3)$ ?

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Answer 1 · 2017-05-18T23:29:04

$f (g (x)) = \sqrt{\sqrt{x + 3} - 3}$ $f(g(x)) = sqrt(sqrt(x + 3) - 3)$

$g (f (x)) = \sqrt{\sqrt{x - 3} + 3}$ $g(f(x)) = sqrt(sqrt(x - 3) + 3)$

The two possible compositions are $f (g (x))$ $f(g(x))$ and $g (f (x))$ $g(f(x))$ :

$\Rightarrow f (g (x)) = f (\sqrt{x + 3})$ $Rightarrow f(g(x)) = f(sqrt(x + 3))$

$\Rightarrow f (g (x)) = \sqrt{\sqrt{x + 3} - 3}$ $Rightarrow f(g(x)) = sqrt(sqrt(x + 3) - 3)$

$and$ $and$

$\Rightarrow g (f (x)) = g (\sqrt{x - 3})$ $Rightarrow g(f(x)) = g(sqrt(x - 3))$

$\Rightarrow g (f (x)) = \sqrt{\sqrt{x - 3} + 3}$ $Rightarrow g(f(x)) = sqrt(sqrt(x - 3) + 3)$

How do you find the compositions given f(x)=√x−3f(x)=sqrt(x-3), g(x)=√x+3g(x)=sqrt(x +3)?