Pre-Algebra Examples

$f (x) = | x | + \frac{1}{x}$

Step 1

Find the domain for $f (x) = | x | + \frac{1}{x}$ so that a list of $x$ values can be picked to find a list of points, which will help graphing the absolute value function.

Tap for more steps...

Step 1.1

Set the denominator in $\frac{1}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 1.2

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Step 2

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.

Tap for more steps...

Step 2.1

Substitute the $x$ value $- 2$ into $f (x) = | x | + \frac{1}{x}$ . In this case, the point is $(- 2, \frac{3}{2})$ .

Tap for more steps...

Step 2.1.1

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

$f (- 2) = | - 2 | + \frac{1}{- 2}$

Step 2.1.2

Simplify the result.

Tap for more steps...

Step 2.1.2.1

Simplify each term.

Tap for more steps...

Step 2.1.2.1.1

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

$f (- 2) = 2 + \frac{1}{- 2}$

Step 2.1.2.1.2

Move the negative in front of the fraction.

$f (- 2) = 2 - \frac{1}{2}$

Step 2.1.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (- 2) = 2 \cdot \frac{2}{2} - \frac{1}{2}$

Step 2.1.2.3

Combine $2$ and $\frac{2}{2}$ .

$f (- 2) = \frac{2 \cdot 2}{2} - \frac{1}{2}$

Step 2.1.2.4

Combine the numerators over the common denominator.

$f (- 2) = \frac{2 \cdot 2 - 1}{2}$

Step 2.1.2.5

Simplify the numerator.

Tap for more steps...

Step 2.1.2.5.1

Multiply $2$ by $2$ .

$f (- 2) = \frac{4 - 1}{2}$

Step 2.1.2.5.2

Subtract $1$ from $4$ .

$f (- 2) = \frac{3}{2}$

Step 2.1.2.6

The final answer is $\frac{3}{2}$ .

$y = \frac{3}{2}$

Step 2.2

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

Tap for more steps...

Step 2.2.1

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

$f (- 1) = | - 1 | + \frac{1}{- 1}$

Step 2.2.2

Simplify the result.

Tap for more steps...

Step 2.2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.2.1.1

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

$f (- 1) = 1 + \frac{1}{- 1}$

Step 2.2.2.1.2

Divide $1$ by $- 1$ .

$f (- 1) = 1 - 1$

Step 2.2.2.2

Subtract $1$ from $1$ .

$f (- 1) = 0$

Step 2.2.2.3

The final answer is $0$ .

$y = 0$

Step 2.3

Substitute the $x$ value $1$ into $f (x) = | x | + \frac{1}{x}$ . In this case, the point is $(1, 2)$ .

Tap for more steps...

Step 2.3.1

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

$f (1) = | 1 | + \frac{1}{1}$

Step 2.3.2

Simplify the result.

Tap for more steps...

Step 2.3.2.1

Simplify each term.

Tap for more steps...

Step 2.3.2.1.1

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

$f (1) = 1 + \frac{1}{1}$

Step 2.3.2.1.2

Divide $1$ by $1$ .

$f (1) = 1 + 1$

Step 2.3.2.2

Add $1$ and $1$ .

$f (1) = 2$

Step 2.3.2.3

The final answer is $2$ .

$y = 2$

Step 2.4

Substitute the $x$ value $2$ into $f (x) = | x | + \frac{1}{x}$ . In this case, the point is $(2, \frac{5}{2})$ .

Tap for more steps...

Step 2.4.1

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

$f (2) = | 2 | + \frac{1}{2}$

Step 2.4.2

Simplify the result.

Tap for more steps...

Step 2.4.2.1

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

$f (2) = 2 + \frac{1}{2}$

Step 2.4.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (2) = 2 \cdot \frac{2}{2} + \frac{1}{2}$

Step 2.4.2.3

Combine $2$ and $\frac{2}{2}$ .

$f (2) = \frac{2 \cdot 2}{2} + \frac{1}{2}$

Step 2.4.2.4

Combine the numerators over the common denominator.

$f (2) = \frac{2 \cdot 2 + 1}{2}$

Step 2.4.2.5

Simplify the numerator.

Tap for more steps...

Step 2.4.2.5.1

Multiply $2$ by $2$ .

$f (2) = \frac{4 + 1}{2}$

Step 2.4.2.5.2

Add $4$ and $1$ .

$f (2) = \frac{5}{2}$

Step 2.4.2.6

The final answer is $\frac{5}{2}$ .

$y = \frac{5}{2}$

Step 2.5

The absolute value can be graphed using the points around the vertex $(- 2, 1.5), (- 1, 0), (1, 2), (2, 2.5)$

$\begin{array}{c} x & y \\ - 2 & 1.5 \\ - 1 & 0 \\ 1 & 2 \\ 2 & 2.5 \end{array}$

Step 3