Let $f (x) = \frac{3}{x - 1}$ $f(x) = 3/(x-1)$ and $g (x) = \frac{2}{x}$ $g(x) = 2/x$ , how do you find each of the compositions?

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Answer 1 · 2016-01-06T15:04:45

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

$g (f (x)) = \frac{2 (x - 1)}{3}$ $g(f(x)) =( 2(x - 1 ))/3$

(a) $f (g (x)) = f (\frac{2}{x})$ $f(g(x)) = f ( 2/x )$

now substitute $\frac{2}{x}$ $2/x$ in for x in f(x)

$\Rightarrow f (\frac{2}{x}) = \frac{3}{(\frac{2}{x}) - 1}$ $rArr f (2/x) = 3/((2/x) - 1)$

can tidy this up by multiplying numerator and denominator by x.

$\Rightarrow \frac{3}{(\frac{2}{x}) - 1} = \frac{3 x}{2 - x}$ $rArr 3/((2/x) -1 ) = (3x)/(2 - x )$

(b) $g (f (x)) = g (\frac{3}{x - 1})$ $g(f(x)) = g ( 3/(x - 1))$

now substitute $\frac{3}{x - 1}$ $3/ (x - 1 )$ in for x in g(x)

$\Rightarrow g (\frac{3}{x - 1}) = \frac{2}{\frac{3}{(x - 1)}}$ $rArr g (3/(x - 1 )) = 2/(3/((x - 1 ))$

now tidy up

$\Rightarrow g (f (x)) = \frac{2 (x - 1)}{3}$ $rArr g(f(x)) =( 2(x - 1 ))/3$

Let f(x) = 3/(x-1)f(x)=3x−1 f(x) = 3/(x-1) and g(x) = 2/xg(x)=2xg(x) = 2/x, how do you find each of the compositions?