Question

How do you write $f\left( x \right)=\left| x+9 \right|$ as a piecewise function?

Answer 1

Here in this question we have been asked to write the given absolute function $f\left( x \right)=\left| x+9 \right|$ in the form of a piecewise function. From the basic concepts of functions we know that the definition of absolute function $f\left( x \right)=\left| x \right|$ is given as f\left( x \right)=\left\\{ \begin{matrix} x & \forall x\ge 0 \\\ -x & \forall x<0 \\\ \end{matrix} \right\\}.

Complete step by step answer:
Now considering from the question we have been asked to write the given absolute function $f\left( x \right)=\left| x+9 \right|$ in the form of a piecewise function.
From the basic concepts of functions we know that the definition of absolute function $f\left( x \right)=\left| x \right|$ is given as f\left( x \right)=\left\\{ \begin{matrix} x & \forall x\ge 0 \\\ -x & \forall x<0 \\\ \end{matrix} \right\\} .
By using this definition we will have f\left( x \right)=\left\\{ \begin{matrix} \left( x+9 \right) & \forall \left( x+9 \right)\ge 0 \\\ -\left( x+9 \right) & \forall \left( x+9 \right)<0 \\\ \end{matrix} \right\\} .
By further simplifying the domain values by performing arithmetic transformations from left hand side to right hand side we will have f\left( x \right)=\left\\{ \begin{matrix} \left( x+9 \right) & \forall x\ge -9 \\\ -\left( x+9 \right) & \forall x<-9 \\\ \end{matrix} \right\\} .
Therefore we can conclude that the piecewise function form of the given absolute function $f\left( x \right)=\left| x+9 \right|$ is given as f\left( x \right)=\left\\{ \begin{matrix} x+9 & \forall x\ge -9 \\\ -x-9 & \forall x<-9 \\\ \end{matrix} \right\\} .

Note:
While answering questions of this type we should be sure with the functions concepts that we are going to apply in the process and the calculations that we are going to perform in between. This is a very simple and easy question and can be answered accurately in a short span of time. Very few mistakes are possible in this type of question. Someone can make a mistake unintentionally during simplifying and write it as f\left( x \right)=\left\\{ \begin{matrix} x+9 & \forall x\ge -9 \\\ -x+9 & \forall x<-9 \\\ \end{matrix} \right\\}
which is a wrong answer clearly. So we should be careful while simplifying.