Find a formula for the composition of given functions

by RoRi

September 1, 2015

Find the composition function $h(x) = (f \circ g)(x) = f(g(x))$ of the functions:

$f(x) = \sqrt{x}, \qquad \text{if } x > 0, \qquad g(x) = x + \sqrt{x} \quad \text{if } x > 0.$

We compute $h(x) = f(g(x))$ ,

$\begin{align*} h(x) &= f(g(x)) \\ &= f(x + \sqrt{x}) \\ &= \sqrt{ x + \sqrt{x}} \\ \end{align*}$

This is valid for all $x > 0$ . (Of course, this formula would generally be valid for $x = 0$ as well, but since the domain of $f$ and $g$ is given as $x > 0$ , the composition is defined only for $x>0$ .)

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Find a formula for the composition of given functions

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