Precalculus Examples

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

Step 1

Set the radicand in $\sqrt{4 - x^{2}}$ greater than or equal to $0$ to find where the expression is defined.

$4 - x^{2} \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $4$ from both sides of the inequality.

$- x^{2} \geq - 4$

Step 2.2

Divide each term in $- x^{2} \geq - 4$ by $- 1$ and simplify.

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Step 2.2.1

Divide each term in $- x^{2} \geq - 4$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} \leq \frac{- 4}{- 1}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} \leq \frac{- 4}{- 1}$

Step 2.2.2.2

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

$x^{2} \leq \frac{- 4}{- 1}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $- 4$ by $- 1$ .

$x^{2} \leq 4$

Step 2.3

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{x^{2}} \leq \sqrt{4}$

Step 2.4

Simplify the equation.

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

Pull terms out from under the radical.

$| x | \leq \sqrt{4}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify $\sqrt{4}$ .

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Step 2.4.2.1.1

Rewrite $4$ as $2^{2}$ .

$| x | \leq \sqrt{2^{2}}$

Step 2.4.2.1.2

Pull terms out from under the radical.

$| x | \leq | 2 |$

Step 2.4.2.1.3

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

$| x | \leq 2$

Step 2.5

Write $| x | \leq 2$ as a piecewise.

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Step 2.5.1

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

$x \geq 0$

Step 2.5.2

In the piece where $x$ is non-negative, remove the absolute value.

$x \leq 2$

Step 2.5.3

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

$x < 0$

Step 2.5.4

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

$- x \leq 2$

Step 2.5.5

Write as a piecewise.

${\begin{cases} x \leq 2 & x \geq 0 \\ - x \leq 2 & x < 0 \end{cases}$

Step 2.6

Find the intersection of $x \leq 2$ and $x \geq 0$ .

$0 \leq x \leq 2$

Step 2.7

Solve $- x \leq 2$ when $x < 0$ .

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Step 2.7.1

Divide each term in $- x \leq 2$ by $- 1$ and simplify.

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Step 2.7.1.1

Divide each term in $- x \leq 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{2}{- 1}$

Step 2.7.1.2

Simplify the left side.

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Step 2.7.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{2}{- 1}$

Step 2.7.1.2.2

Divide $x$ by $1$ .

$x \geq \frac{2}{- 1}$

Step 2.7.1.3

Simplify the right side.

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Step 2.7.1.3.1

Divide $2$ by $- 1$ .

$x \geq - 2$

Step 2.7.2

Find the intersection of $x \geq - 2$ and $x < 0$ .

$- 2 \leq x < 0$

Step 2.8

Find the union of the solutions.

$- 2 \leq x \leq 2$

Step 3

Set the radicand in $\sqrt{1 - x^{2}}$ greater than or equal to $0$ to find where the expression is defined.

$1 - x^{2} \geq 0$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $1$ from both sides of the inequality.

$- x^{2} \geq - 1$

Step 4.2

Divide each term in $- x^{2} \geq - 1$ by $- 1$ and simplify.

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Step 4.2.1

Divide each term in $- x^{2} \geq - 1$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} \leq \frac{- 1}{- 1}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} \leq \frac{- 1}{- 1}$

Step 4.2.2.2

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

$x^{2} \leq \frac{- 1}{- 1}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $- 1$ by $- 1$ .

$x^{2} \leq 1$

Step 4.3

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{x^{2}} \leq \sqrt{1}$

Step 4.4

Simplify the equation.

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Step 4.4.1

Simplify the left side.

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Step 4.4.1.1

Pull terms out from under the radical.

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

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Any root of $1$ is $1$ .

$| x | \leq 1$

Step 4.5

Write $| x | \leq 1$ as a piecewise.

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Step 4.5.1

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

$x \geq 0$

Step 4.5.2

In the piece where $x$ is non-negative, remove the absolute value.

$x \leq 1$

Step 4.5.3

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

$x < 0$

Step 4.5.4

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

$- x \leq 1$

Step 4.5.5

Write as a piecewise.

${\begin{cases} x \leq 1 & x \geq 0 \\ - x \leq 1 & x < 0 \end{cases}$

Step 4.6

Find the intersection of $x \leq 1$ and $x \geq 0$ .

$0 \leq x \leq 1$

Step 4.7

Solve $- x \leq 1$ when $x < 0$ .

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Step 4.7.1

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

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Step 4.7.1.1

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

$\frac{- x}{- 1} \geq \frac{1}{- 1}$

Step 4.7.1.2

Simplify the left side.

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Step 4.7.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{1}{- 1}$

Step 4.7.1.2.2

Divide $x$ by $1$ .

$x \geq \frac{1}{- 1}$

Step 4.7.1.3

Simplify the right side.

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Step 4.7.1.3.1

Divide $1$ by $- 1$ .

$x \geq - 1$

Step 4.7.2

Find the intersection of $x \geq - 1$ and $x < 0$ .

$- 1 \leq x < 0$

Step 4.8

Find the union of the solutions.

$- 1 \leq x \leq 1$

Step 5

Set the denominator in $\frac{\sqrt{4 - x^{2}}}{\sqrt{1 - x^{2}}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{1 - x^{2}} = 0$

Step 6

Solve for $x$ .

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Step 6.1

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

${\sqrt{1 - x^{2}}}^{2} = 0^{2}$

Step 6.2

Simplify each side of the equation.

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Step 6.2.1

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

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

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

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

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Step 6.2.2.1.1

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

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Step 6.2.2.1.1.1

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

${(1 - x^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 6.2.2.1.1.2.1

Cancel the common factor.

${(1 - x^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.2.2.1.1.2.2

Rewrite the expression.

${(1 - x^{2})}^{1} = 0^{2}$

Step 6.2.2.1.2

Simplify.

$1 - x^{2} = 0^{2}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Raising $0$ to any positive power yields $0$ .

$1 - x^{2} = 0$

Step 6.3

Solve for $x$ .

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Step 6.3.1

Subtract $1$ from both sides of the equation.

$- x^{2} = - 1$

Step 6.3.2

Divide each term in $- x^{2} = - 1$ by $- 1$ and simplify.

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Step 6.3.2.1

Divide each term in $- x^{2} = - 1$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 1}{- 1}$

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 1}{- 1}$

Step 6.3.2.2.2

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

$x^{2} = \frac{- 1}{- 1}$

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

Divide $- 1$ by $- 1$ .

$x^{2} = 1$

Step 6.3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{1}$

Step 6.3.4

Any root of $1$ is $1$ .

$x = \pm 1$

Step 6.3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 6.3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 1$

Step 6.3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 1$

Step 6.3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 1, - 1$

Step 7

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

Interval Notation:

$(- 1, 1)$

Set-Builder Notation:

${x | - 1 < x < 1}$

Step 8