Question

How do you create a table of values for an absolute value function?

Answer 1

Hint: We recall the definition of absolute value function $\left| x \right|$ which returns positive outputs irrespective of whether the input is positive, negative or zero. We take different positive, negative and zero inputs $x$ in the first column and corresponding positive outputs $\left| x \right|$ in the second column to draw the table for the equation $y=\left| x \right|$.

Complete step-by-step solution:
We know that every function takes inputs and returns outputs such that no one input gives two outputs. A table for functional values is drawn with inputs in the first column and outputs in the second column. \[\]
We know that absolute value function or modulus function makes a number positive and it is denoted as $\left| x \right|$. If $\left| x \right|$ takes 1 as an input it will return 1, if $\left| x \right|$ takes $-1$ as an input it will return 1.
Now we think of how to make a rule for all negative numbers such that they get converted to positive numbers, the answer is we multiply $-1$ to each negative number. So if $x$ is a positive number $\left| x \right|$ will return $x$ and if $x$ is negative number $\left| x \right|$ will return $-1\times x=-x$. Now the only number that is neither negative nor positive is zero and the absolute function returns 0 if the input is zero. So the function is defined in a piecewise manner as
\[\left| x \right|=\left\{ \begin{matrix}
   x, & \text{if }x\ge 0\text{ } \\
   -x, & \text{if }x < 0 \\
\end{matrix} \right.\]
So let take us numbers $-3,-2,-1,0,-1,2,3$ as inputs and find the outputs as below,
\[\begin{align}
  & \left| -2 \right|=-\left( -2 \right)=2 \\
 & \left| -1 \right|=-\left( -1 \right)=1 \\
 & \left| -0 \right|=0 \\
 & \left| 1 \right|=1 \\
 & \left| 2 \right|=2 \\
\end{align}\]
We draw the table for absolute value function below.\[\]

x	$y= \left| x \right|$
-2	2
-1	1
0	0
1	1
2	2

Note: We note that the function which takes inputs is called domain and the set to which it returns values is called range. The domain of absolute value function is real number set and range is non-negative real number set. Another type of absolute value equation is $y=\left| x-a \right|$ where $y=-\left( x-a \right)$ if $x < -a$ and $y=x-a$ if $y\ge a$.