Tag: Composition

Find a formula for the composition of given functions

by RoRi

August 31, 2015

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

$f(x) = \sqrt{x} \quad \text{if } x \geq 0, \qquad g(x) = -x^2.$

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

$\begin{align*} h(x) &= f(g(x)) \\ &= f(-x^2) \\ &= \sqrt{-x^2} \end{align*}$

Since $-x^2 < 0$ if $x \neq 0$ and $f$ is undefined for any $x < 0$ , we have $h(x) = 0$ if $x = 0$ , and $h$ is undefined for all $x \neq 0$ .

Find a formula for the composition of given functions

by RoRi

August 31, 2015

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

$f(x) = \sqrt{x} \quad \text{if } x \geq 0, \qquad g(x) = x^2.$

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

$\begin{align*} h(x) &= f(g(x)) \\ &= f(x^2) \\ &= \sqrt{x^2} \\ &= |x| \end{align*}$

This is valid for all $x \in \mathbb{R}$ . (Although $f$ only takes non-negative $x$ , it is not a problem since $g(x) \geq 0$ for any $x \in \mathbb{R}$ .)

Find a formula for the composition of given functions

by RoRi

August 31, 2015

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

$f(x) = x+1, \qquad g(x) = x^2 -2x.$

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

$\begin{align*} h(x) &= f(g(x)) \\ &= f(x^2 - 2x) \\ &= (x^2 - 2x) + 1 \\ &= (x-1)^2 \end{align*}$

This is valid for all $x \in \mathbb{R}$ .

Find a formula for the composition of given functions

by RoRi

August 31, 2015

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

$f(x) = x^2 - 2x, \qquad g(x) = x + 1.$

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

$\begin{align*} h(x) &= f(g(x)) \\ &= f(x+1) \\ &= (x+1)^2 - 2 (x+1) \\ &= x^2 + 2x + 1 - 2x - 2 \\ &= x^2 - 1. \end{align*}$

This is valid for all $x \in \mathbb{R}$ .