How do you find the compositions given $f (x) = 2 - x$ $f(x)=2-x$ and $g (x) = \frac{2}{5 - x}$ $g(x)=2/(5-x)$ ?

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How do you find the compositions given $f (x) = 2 - x$ $f(x)=2-x$ and $g (x) = \frac{2}{5 - x}$ $g(x)=2/(5-x)$ ?

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Answer 1 · 2018-07-19T23:33:02

There are two possible compositions: $f (g (x))$ $f(g(x))$ and $g (f (x))$ $g(f(x))$ .
We can do either:

$f (g (x)) = f (\frac{2}{5 - x}) = 2 - \frac{2}{5 - x} = \frac{8 - 2 x}{5 - x}$ $f(g(x)) = f(2/(5-x)) = 2 - 2/(5-x) = (8-2x)/(5-x)$
$g (f (x)) = g (2 - x) = \frac{2}{5 - (2 - x)} = \frac{2}{3 + x}$ $g(f(x)) = g(2-x) = 2/(5-(2-x)) = 2/(3+x)$

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How do you find the compositions given $f (x) = 2 - x$ $f(x)=2-x$ and $g (x) = \frac{2}{5 - x}$ $g(x)=2/(5-x)$ ?

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