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.

47.

$\begin{array}{l}f\left(x\right)=2x+3\\ g\left(x\right)=4x-1\end{array}$

Short answer

$\begin{array}{l}f\circ g=8x+1\\ g\circ f=8x+11\\ f\circ f=4x+9\\ g\circ g=16x-5\end{array}$

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

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

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

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

Step-by-step solution

Step 1- Given Information

Given that,

$\begin{array}{l}f\left(x\right)=2x+3\\ g\left(x\right)=4x-1\end{array}$

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

$\begin{array}{l}f\circ g=f\left(g\left(x\right)\right)\\ \text{=}f\left(4x-1\right)\\ \text{=2}\left(4x-1\right)+3\\ \text{=2}\left(4x\right)-2\left(1\right)+3\\ \text{=}8x-2+3\\ \text{=}8x+1\\ \Rightarrow f\circ g=8x+1\end{array}$

Also, the domain $f\circ g$is all real numbers$\left(-\infty ,\infty \right)$

Step 4- To find g∘f

By using Step 2, we have

$\begin{array}{l}g\circ f=g\left(f\left(x\right)\right)\\ \text{=g}\left(2x+3\right)\\ \text{=4}\left(2x+3\right)-1\\ \text{=4}\left(2x\right)+4\left(3\right)-1\\ \text{=}8x+12-1\\ \text{=}8x+11\\ \Rightarrow g\circ f=8x+11\end{array}$

Also, the domain $g\circ f$is all real numbers$\left(-\infty ,\infty \right)$

Step 5- To find f∘f

By using Step 2, we have

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

Also, the domain $f\circ f$is all real numbers$\left(-\infty ,\infty \right)$

Step 6- To find g∘g

By using Step 2, we have

$\begin{array}{l}g\circ g=g\left(g\left(x\right)\right)\\ \text{=g}\left(4x-1\right)\\ \text{=4}\left(4x-1\right)-1\\ \text{=4}\left(4x\right)-4\left(1\right)-1\\ \text{=16}x-4-1\\ \text{=16}x-5\\ \Rightarrow g\circ g=\text{16}x-5\end{array}$

Also, the domain $g\circ g$is all real numbers$\left(-\infty ,\infty \right)$