Question

Let f:R---->R and g:R--->R be defined by f(x)=x3 - 4x , g(x)=1/x2 + 1 and h(x)=x4 Find I. (fogoh)(x) II. (hogof)(x) III. (gog)(x) IV. (goh)(x)

Answer 1

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Q1) Let f:R....>R and g:R....>R be defined by f(x)=x³-4x ,g(x)=1/x²+1

and h(x)=x*4 :

$\: \: \boxed { \bf \purple{to \: find \to \: (1) \: f(g(h(x)) \: (2) \: h(g(f(x)) \: and \: (3) \: g(g(x)) \: and \: (4) \: g(h(x))}}$

$\pink \bigstar\bf{(1) \: f(g(h(x) \to}$

$\implies \bf{f(g(h(x)) = f(g( {x}^{4} ) = f( \frac{1}{ {x}^{6} + 1 } ) }$

$\implies \bf{f( \frac{1}{ {x}^{6} + 1 } ) = {( \frac{1}{ {x}^{6} + 1 }) }^{3} - 4( \frac{1}{ {x}^{6} + 1 } )}$

$\red \bigstar \bf{(2) \: h(g(f(x)) \to}$

$\implies \bf{h(g( {x}^{3} - 4x)) = h( \frac{1}{ \ ({ {x}^{3} - 4x)}^{2} + 1 } })$

$\implies \bf{ (\frac{1}{ { ({x}^{3} - 4x) }^{2} + 1} )^{4} }$