Find $gof$ and $fog$,if $f: R \rightarrow R$ and $g: R \rightarrow R$ are given by $f(x) = \cos x$ and $g(x) = 3x^2$. Show that $gof \neq fog$.
(N/A) Given $f(x) = \cos x$ and $g(x) = 3x^2$.
First,we find $gof(x) = g(f(x)) = g(\cos x) = 3(\cos x)^2 = 3 \cos^2 x$.
Next,we find $fog(x) = f(g(x)) = f(3x^2) = \cos(3x^2)$.
To show $gof \neq fog$,we can test a value,for example,$x = 0$.
$gof(0) = 3 \cos^2(0) = 3(1)^2 = 3$.
$fog(0) = \cos(3(0)^2) = \cos(0) = 1$.
Since $3 \neq 1$,it follows that $gof \neq fog$.
First,we find $gof(x) = g(f(x)) = g(\cos x) = 3(\cos x)^2 = 3 \cos^2 x$.
Next,we find $fog(x) = f(g(x)) = f(3x^2) = \cos(3x^2)$.
To show $gof \neq fog$,we can test a value,for example,$x = 0$.
$gof(0) = 3 \cos^2(0) = 3(1)^2 = 3$.
$fog(0) = \cos(3(0)^2) = \cos(0) = 1$.
Since $3 \neq 1$,it follows that $gof \neq fog$.
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