Question

Let f : [0, ∞) → R and g : R → R be defined by $f\left(x\right)=\sqrt{x}$and g(x) = x. Find f + g, f − g, fg and $\frac{f}{g}$.

Answer 1

It is given that f : [0, ∞) → R and g : R → R such that $f\left(x\right)=\sqrt{x}$ and g(x) = x .
$D\left(f+g\right)=[0,\infty )\cap R=[0,\infty )$
So, f + g : [0, ∞) → R is given by
$\left(f+g\right)\left(x\right)=f\left(x\right)+g\left(x\right)=\sqrt{x}+x$

$D\left(f-g\right)=D\left(f\right)\cap D\left(g\right)=[0,\infty )\cap R=[0,\infty )$
So, f - g : [0, ∞) → R is given by
$\left(f-g\right)\left(x\right)=f\left(x\right)-g\left(x\right)=\sqrt{x}-x$

$D\left(fg\right)=D\left(f\right)\cap D\left(g\right)=[0,\infty )\cap R=[0,\infty )$
So, fg : [0, ∞) → R is given by
$\left(fg\right)\left(x\right)=f\left(x\right)g\left(x\right)=\sqrt{x}.x=x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

$D\left(\frac{f}{g}\right)=\left[D\left(f\right)\cap D\left(g\right)-\left\{x:g\left(x\right)=0\right\}\right]=\left(0,\infty \right)$
So,$\frac{f}{g}:\left(0,\infty \right)\to R$ is given by
$\left(\frac{f}{g}\right)\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}=\frac{\sqrt{x}}{x}=\frac{1}{\sqrt{x}}$