Find points at which a given composite function is continuous

by RoRi

September 4, 2015

Define functions $f$ and $g$ as follows:

$f(x) = \frac{x+|x|}{2}, \text{ for all } x, \qquad g(x) = \begin{cases} x & \text{for } x < 0,\\ x^2 &\text{for } x \geq 0. \end{cases}$

Find a formula for $h(x) = g(f(x))$ . Find the values at which $h(x)$ is continuous.

If $x < 0$ , we have $|x| = -x$ and we compute the composition,

$h(x) = g(f(x)) = g \left( \frac{x - x}{2} \right) = g(0) = 0.$

If $x \geq 0$ , then $|x| = x$ and we compute the composition,

$h(x) = g(f(x)) = g \left( \frac{x+x}{2} \right) = g(x) = x^2.$

Hence,

$h(x) = \begin{cases} 0 & \text{if } x < 0, \\ x^2 & \text{if } x \geq 0. \end{cases}$

This is the same function as in this previous exercise, and it is continuous everywhere.

Stumbling Robot