Algebra Examples

$x = - \sqrt{y}$

Rewrite the equation as $- \sqrt{y} = x$ .

$- \sqrt{y} = x$

Divide each term in $- \sqrt{y} = x$ by $- 1$ and simplify.

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Divide each term in $- \sqrt{y} = x$ by $- 1$ .

$\frac{- \sqrt{y}}{- 1} = \frac{x}{- 1}$

Simplify the left side.

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Dividing two negative values results in a positive value.

$\frac{\sqrt{y}}{1} = \frac{x}{- 1}$

Divide $\sqrt{y}$ by $1$ .

$\sqrt{y} = \frac{x}{- 1}$

Simplify the right side.

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Move the negative one from the denominator of $\frac{x}{- 1}$ .

$\sqrt{y} = - 1 \cdot x$

Rewrite $- 1 \cdot x$ as $- x$ .

$\sqrt{y} = - x$

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{y}}^{2} = {(- x)}^{2}$

Simplify each side of the equation.

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

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

Simplify the left side.

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Simplify ${(y^{\frac{1}{2}})}^{2}$ .

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Multiply the exponents in ${(y^{\frac{1}{2}})}^{2}$ .

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

$y^{\frac{1}{2} \cdot 2} = {(- x)}^{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$y^{\frac{1}{2} \cdot 2} = {(- x)}^{2}$

Rewrite the expression.

$y^{1} = {(- x)}^{2}$

Simplify.

$y = {(- x)}^{2}$

Simplify the right side.

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Simplify ${(- x)}^{2}$ .

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Apply the product rule to $- x$ .

$y = {(- 1)}^{2} x^{2}$

Raise $- 1$ to the power of $2$ .

$y = 1 x^{2}$

Multiply $x^{2}$ by $1$ .

$y = x^{2}$

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:

${x | x \in ℝ}$

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${y | y \geq 0}$

Determine the domain and range.

Domain: $(- \infty, \infty), {x | x \in ℝ}$

Range: $[0, \infty), {y | y \geq 0}$