Question

In Problems 53–58, find functionsf and g so that $f\circ g=H$.

$H\left(x\right)=\left|2x+1\right|$

Short answer

The required functions are$f\left(x\right)=\left|x\right|;g\left(x\right)=2x+1$.

Step-by-step solution

Step 1. Given information.

The given function is:

$H\left(x\right)=\left|2x+1\right|$

In the given function H takes $2x+1$and takes the absolute value of the function.

Now decompose H by taking the absolute value of the function.

Let's take $g\left(x\right)=2x+1$and $f\left(x\right)=\left|x\right|$.

Step 2. Find f∘g.

$(f\circ g)\left(x\right)=f\left(g\right(x\left)\right)$

Substitute$g\left(x\right)=2x+1$in the function $f\left(g\right(x\left)\right)$,

Then the function will become $f(2x+1)$.

Now replace x with $2x+1$in $f\left(x\right)=\left|x\right|$,

$$\begin{gathered} f(2x+1)=\left|2x+1\right| \\ \left|2x+1\right|=H\left(x\right) \end{gathered}$$

As we can see that$f\circ g=H$, therefore the values of the function that we assumed are correct.

$f\left(x\right)=\left|x\right|;g\left(x\right)=2x+1$