How do you compute (fog) and (gof) if $f(x)= x/(x-2)$ , $g(x)=3/x$ ?

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Answer 1 · 2016-01-02T08:16:08

$(f@g)(x)=3/(3-2x),(g@f)(x)=(3(x-2))/x$

$(f@g)(x)$ is the same as $f(g(x))$ . This means that you take $g(x)$ , or $3/x$ , plug it in for all the spots with $x$ in $f(x)$ .

$f(g(x))=(3/x)/(3/x-2)$

Find a common denominator in the denominator.

$f(g(x))=(3/x)/(3/x-(2x)/x)$

Simplify.

$f(g(x))=(3/x)/((3-2x)/x)$

Multiply by $x/x$ .

$f(g(x))=3/(3-2x)=(f@g)(x)$

To find $(g@f)(x)$ , use a similar method: plug $x/(x-2)$ into the $x$ in $3/x$ .

$g(f(x))=3/(x/(x-2))$

Recall that division is the same as multiplying by the reciprocal.

$g(f(x))=3((x-2)/x)$

Simplify for the final answer.

$g(f(x))=(3(x-2))/x=(g@f)(x)$

How do you compute (fog) and (gof) if f(x)= x/(x-2)f(x)= x/(x-2), g(x)=3/xg(x)=3/x?