Question
If $$f\left( x \right) = {x^n},\,n\, \in \,N$$ and $$\left( {g\,o\,f} \right)\left( x \right) = ng\left( x \right)$$ then $$g\left( x \right)$$ can be :
A.
$$n\,\left| x \right|$$
B.
$$3 \cdot \root 3 \of x $$
C.
$${e^x}$$
D.
$$\log \,\left| x \right|$$
Answer :
$$\log \,\left| x \right|$$
Solution :
$$g\left\{ {f\left( x \right)} \right\} = g\left( {{x^n}} \right) = ng\left( x \right).$$ Also $$\log \,{x^n} = n\,\log \,\left| x \right|.$$ So, $$g\left( x \right) = \log \,\left| x \right|$$ is possible.