Answer to Question #237604 in Calculus for Anand

Question

Express the function g, defined by g(x)=1/√(x+√x) as a composition of three functions.

Answer 1

Solution;

One possible way ;

If you let;

$f(x)=\sqrt{x}$

$g(x)=\sqrt{x^2+x}$

$h(x)=\frac1x$

First,obtain g of f(x);

$g o f=g(f(x))=\sqrt{(f(x))^2+f(x)}=\sqrt{(\sqrt{x})^2+\sqrt{x}}=\sqrt{x+\sqrt{x}}$

Now we plug g(f(x)) into h(x);

$F(x)=h (gof)=h(g(f(x)))$

$F(x)=\frac{1}{g(f(x))}$ =$\frac{1}{\sqrt{x+\sqrt{x}}}$

Hence the expression of the function is three functions could be expressed as;

$f(x)=\sqrt{x}$ ,$g(x)=\sqrt{x^2+x}$ , $h(x)=\frac1x$ ,$F(x)=h$ o $g$ o $f$