Question

Composition of functions- Find the functions$f\circ g,g\circ f,f\circ f,\text{\hspace{0.33em}and}g\circ g$and their domains.

width="97">fx=xgx=2x+3

Short answer

$\begin{array}{l}f\circ g=\left|2x+3\right|\\ g\circ f=2\left|x\right|+3\\ f\circ f=\left|\left|x\right|\right|\\ g\circ g=4x+9\end{array}$

The domain$f\circ g$ is all real numbers i.e.,$\left(-\infty ,\infty \right)$

The domain$g\circ f$ is all real numbers i.e.,$\left(-\infty ,\infty \right)$

The domain$f\circ f$ is all real numbers i.e.,$\left(-\infty ,\infty \right)$

The domain $g\circ g$is all real numbers i.e.,$\left(-\infty ,\infty \right)$

Step-by-step solution

Step 1. Given Information.

Given that,

$\begin{array}{l}f\left(x\right)=\left|x\right|\\ g\left(x\right)=2x+3\end{array}$

Step 2. Concept used.

The Compositions are defined as

$\begin{array}{l}f\circ g=f\left(g\left(x\right)\right)\\ g\circ f=g\left(f\left(x\right)\right)\\ f\circ f=f\left(f\left(x\right)\right)\\ g\circ g=g\left(g\left(x\right)\right)\end{array}$

Step 3. To find f∘g.

By using Step 2, we have

$\begin{array}{l}f\circ g=f\left(g\left(x\right)\right)\\ \text{=}f\left(2x+3\right)\\ \text{=}\left|2x+3\right|\\ \Rightarrow f\circ g=\left|2x+3\right|\end{array}$

And

The domain of $f\circ g$is all real numbers i.e.,$\left(-\infty ,\infty \right)$

Step 4. To find g∘f.

By using Step 2, we have

$\begin{array}{l}g\circ f=g\left(f\left(x\right)\right)\\ \text{=g}\left(\left|x\right|\right)\\ \text{=}2\left|x\right|+3\\ \Rightarrow g\circ f=2\left|x\right|+3\end{array}$

So,

The domain of $g\circ f$is all real numbers i.e., width="61">−∞,∞

Step 5. To find f∘f.

By using Step 2, we have

$\begin{array}{l}f\circ f=f\left(f\left(x\right)\right)\\ \text{=}f\left(\left|x\right|\right)\\ \text{=}\left|\left|x\right|\right|\\ \Rightarrow f\circ f=\left|\left|x\right|\right|\end{array}$

Also, the domain $f\circ f$is all real numbers i.e.,width="61">−∞,∞

Step 6. To find g∘g.

By using Step 2, we have

$\begin{array}{l}g\circ g=g\left(g\left(x\right)\right)\\ \text{=g}\left(2x+3\right)\\ \text{=}2\left(2x+3\right)+3\\ \text{=}2\left(2x\right)+2\left(3\right)+3\\ \text{=}4x+6+3\\ \text{=}4x+9\\ \Rightarrow g\circ g=4x+9\end{array}$

So,

The domain $g\circ g$is all real numbers i.e.,$\left(-\infty ,\infty \right)$