Graph x-|x|

$x - | x |$

Step 1

Find the absolute value vertex. In this case, the vertex for $y = x - | x |$ is $(0, 0)$ .

To find the $x$ coordinate of the vertex, set the inside of the absolute value $x$ equal to $0$ . In this case, $x = 0$ .

$x = 0$

Replace the variable $x$ with $0$ in the expression.

$y = (0) - | 0 |$

Simplify $(0) - | 0 |$ .

Simplify each term.

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$y = 0 - 0$

Multiply $- 1$ by $0$ .

$y = 0 + 0$

Add $0$ and $0$ .

$y = 0$

The absolute value vertex is $(0, 0)$ .

$(0, 0)$

Step 2

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 3

For each $x$ value, there is one $y$ value. Select a few $x$ values from the domain. It would be more useful to select the values so that they are around the $x$ value of the absolute value vertex.

Substitute the $x$ value $- 2$ into $f (x) = x - | x |$ . In this case, the point is $(- 2, - 4)$ .

Replace the variable $x$ with $- 2$ in the expression.

$f (- 2) = (- 2) - | - 2 |$

Simplify the result.

Simplify each term.

The absolute value is the distance between a number and zero. The distance between $- 2$ and $0$ is $2$ .

$f (- 2) = - 2 - 1 \cdot 2$

Multiply $- 1$ by $2$ .

$f (- 2) = - 2 - 2$

Subtract $2$ from $- 2$ .

$f (- 2) = - 4$

The final answer is $- 4$ .

$y = - 4$

Substitute the $x$ value $- 1$ into $f (x) = x - | x |$ . In this case, the point is $(- 1, - 2)$ .

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = (- 1) - | - 1 |$

Simplify the result.

Simplify each term.

The absolute value is the distance between a number and zero. The distance between $- 1$ and $0$ is $1$ .

$f (- 1) = - 1 - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$f (- 1) = - 1 - 1$

Subtract $1$ from $- 1$ .

$f (- 1) = - 2$

The final answer is $- 2$ .

$y = - 2$

Substitute the $x$ value $1$ into $f (x) = x - | x |$ . In this case, the point is $(1, 0)$ .

Replace the variable $x$ with $1$ in the expression.

$f (1) = (1) - | 1 |$

Simplify the result.

Simplify each term.

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$f (1) = 1 - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$f (1) = 1 - 1$

Subtract $1$ from $1$ .

$f (1) = 0$

The final answer is $0$ .

$y = 0$

The absolute value can be graphed using the points around the vertex $(0, 0), (- 2, - 4), (- 1, - 2), (1, 0), (2, 0)$

$\begin{array}{c} x & y \\ - 2 & - 4 \\ - 1 & - 2 \\ 0 & 0 \\ 1 & 0 \\ 2 & 0 \end{array}$

Step 4

Graph x-|x|

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