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 

\[\left( fg \right)\left( x \right) = f\left( x \right)g\left( x \right) = \sqrt{x} . x = x^\frac{3}{2}\]

\[D\left( \frac{f}{g} \right) = \left[ D\left( f \right) \cap D\left( g \right) - \left\{ x: g\left( x \right) = 0 \right\} \right] = \left( 0, \infty \right)\]

So,

\[\frac{f}{g}: \left( 0, \infty \right) \to R\]  is given by

\[\left( \frac{f}{g} \right)\left( x \right) = \frac{f\left( x \right)}{g\left( x \right)} = \frac{\sqrt{x}}{x} = \frac{1}{\sqrt{x}}\]