Algebra Examples

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

Step 1

Find the absolute value vertex. In this case, the vertex for $y = {| x |}^{\frac{1}{2}}$ is $(0, 0)$ .

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Step 1.1

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

$x = 0$

Step 1.2

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

$y = {| 0 |}^{\frac{1}{2}}$

Step 1.3

Simplify ${| 0 |}^{\frac{1}{2}}$ .

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Step 1.3.1

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

$y = 0^{\frac{1}{2}}$

Step 1.3.2

Simplify the expression.

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Step 1.3.2.1

Rewrite $0$ as $0^{2}$ .

$y = {(0^{2})}^{\frac{1}{2}}$

Step 1.3.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = 0^{2 (\frac{1}{2})}$

Step 1.3.3

Cancel the common factor of $2$ .

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Step 1.3.3.1

Cancel the common factor.

$y = 0^{2 (\frac{1}{2})}$

Step 1.3.3.2

Rewrite the expression.

$y = 0$

Step 1.3.4

Evaluate the exponent.

$y = 0$

Step 1.4

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

$(0, 0)$

Step 2

Find the domain for $f (x) = {| x |}^{\frac{1}{2}}$ so that a list of $x$ values can be picked to find a list of points, which will help graphing the absolute value function.

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Step 2.1

Convert expressions with fractional exponents to radicals.

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Step 2.1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\sqrt{{| x |}^{1}}$

Step 2.1.2

Anything raised to $1$ is the base itself.

$\sqrt{| x |}$

Step 2.2

Set the radicand in $\sqrt{| x |}$ greater than or equal to $0$ to find where the expression is defined.

$| x | \geq 0$

Step 2.3

Solve for $x$ .

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Step 2.3.1

Write $| x | \geq 0$ as a piecewise.

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Step 2.3.1.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 2.3.1.2

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 2.3.1.3

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x \geq 0$

Step 2.3.1.4

Write as a piecewise.

${\begin{cases} x \geq 0 & x \geq 0 \\ - x \geq 0 & x < 0 \end{cases}$

Step 2.3.2

Find the intersection of $x \geq 0$ and $x \geq 0$ .

$x \geq 0$

Step 2.3.3

Solve $- x \geq 0$ when $x < 0$ .

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Step 2.3.3.1

Divide each term in $- x \geq 0$ by $- 1$ and simplify.

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Step 2.3.3.1.1

Divide each term in $- x \geq 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{0}{- 1}$

Step 2.3.3.1.2

Simplify the left side.

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Step 2.3.3.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{0}{- 1}$

Step 2.3.3.1.2.2

Divide $x$ by $1$ .

$x \leq \frac{0}{- 1}$

Step 2.3.3.1.3

Simplify the right side.

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Step 2.3.3.1.3.1

Divide $0$ by $- 1$ .

$x \leq 0$

Step 2.3.3.2

Find the intersection of $x \leq 0$ and $x < 0$ .

$x < 0$

Step 2.3.4

Find the union of the solutions.

All real numbers

Step 2.4

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

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.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

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 3.1.2.2

The final answer is $2^{\frac{1}{2}}$ .

$y = 2^{\frac{1}{2}}$

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Simplify the result.

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Step 3.2.2.1

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

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

Step 3.2.2.2

One to any power is one.

$f (- 1) = 1$

Step 3.2.2.3

The final answer is $1$ .

$y = 1$

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

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 3.3.2.2

The final answer is $2^{\frac{1}{2}}$ .

$y = 2^{\frac{1}{2}}$

Step 3.4

The absolute value can be graphed using the points around the vertex $(0, 0), (- 2, 1.41), (- 1, 1), (1, 1), (2, 1.41)$

$\begin{array}{c} x & y \\ - 2 & 1.41 \\ - 1 & 1 \\ 0 & 0 \\ 1 & 1 \\ 2 & 1.41 \end{array}$

Step 4