Question

How do you write $y=\left| 6+2x \right|+1$ as a piecewise function?

Answer 1

Hint: A piecewise is a function which is defined differently in different intervals within its domain. So we have to write the different definitions of the given function in the different intervals of x, which can be determined by using the definition of the modulus function. The modulus function returns the absolute value of the argument. So we have to consider two cases, one for $6+2x\ge 0$ and the other for $6+2x<0$.

Complete step-by-step solution:
The function given in the above question is
$\Rightarrow y=\left| 6+2x \right|+1......\left( i \right)$
We know that the modulus function returns the absolute value of the argument. Therefore, we can consider the two cases as below.
Case I: When $6+2x\ge 0$
$\Rightarrow 6+2x\ge 0$
Subtracting $6$ from both sides
$\begin{align}
  & \Rightarrow 6+2x-6\ge 0-6 \\
 & \Rightarrow 2x\ge -6 \\
\end{align}$
Dividing both sides by $2$
$\begin{align}
  & \Rightarrow \dfrac{2x}{2}\ge \dfrac{-6}{2} \\
 & \Rightarrow x\ge -3 \\
\end{align}$
Since in this case, the argument of the modulus function is non-negative, its absolute value will be equal to the argument. Therefore for $x\ge -3$ we can write
$\Rightarrow \left| 6+2x \right|=6+2x$
Substituting this in the equation (i) we get
$\begin{align}
  & \Rightarrow y=6+2x+1 \\
 & \Rightarrow y=2x+7 \\
\end{align}$
Case II: When $6+2x\ge 0$
$\Rightarrow 6+2x<0$
Subtracting $6$ from both sides
$\begin{align}
  & \Rightarrow 6+2x-6<0-6 \\
 & \Rightarrow 2x<-6 \\
\end{align}$
Dividing both sides by $2$
$\begin{align}
  & \Rightarrow \dfrac{2x}{2}<\dfrac{-6}{2} \\
 & \Rightarrow x<-3 \\
\end{align}$
Since in this case, the argument of the modulus function is non-negative, its absolute value will be equal to the argument. Therefore for $x<-3$ we can write
\[\Rightarrow \left| 6+2x \right|=-\left( 6+2x \right)\]
Substituting this in the equation (i) we get
\[\begin{align}
  & \Rightarrow y=-\left( 6+2x \right)+1 \\
 & \Rightarrow y=-6-2x+1 \\
 & \Rightarrow y=-2x-5 \\
\end{align}\]
From the above two cases, we can write the given function in the piecewise form as
\[\Rightarrow y=\left\{ \begin{align}
  & 2x+7;x\ge -3 \\
 & -2x-5;x<-3 \\
\end{align} \right\}\]
We can see these two pieces of the given function in the below graph.

Hence, we have written the given function in the piecewise form.


Note: Whenever a function is defined in terms of modulus, greatest integer, fractional part etc. it is written in the piecewise form, since these have different definitions for the different intervals. We can put the equality sign on both of the inequalities for two intervals also.