Precalculus Examples

$\frac{t}{\sqrt{t^{2} - 45}}$

Step 1

Set the radicand in $\sqrt{t^{2} - 45}$ greater than or equal to $0$ to find where the expression is defined.

$t^{2} - 45 \geq 0$

Step 2

Solve for $t$ .

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Step 2.1

Add $45$ to both sides of the inequality.

$t^{2} \geq 45$

Step 2.2

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

$\sqrt{t^{2}} \geq \sqrt{45}$

Step 2.3

Simplify the equation.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Pull terms out from under the radical.

$| t | \geq \sqrt{45}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify $\sqrt{45}$ .

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Step 2.3.2.1.1

Rewrite $45$ as $3^{2} \cdot 5$ .

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Step 2.3.2.1.1.1

Factor $9$ out of $45$ .

$| t | \geq \sqrt{9 (5)}$

Step 2.3.2.1.1.2

Rewrite $9$ as $3^{2}$ .

$| t | \geq \sqrt{3^{2} \cdot 5}$

Step 2.3.2.1.2

Pull terms out from under the radical.

$| t | \geq | 3 | \sqrt{5}$

Step 2.3.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$| t | \geq 3 \sqrt{5}$

Step 2.4

Write $| t | \geq 3 \sqrt{5}$ as a piecewise.

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Step 2.4.1

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

$t \geq 0$

Step 2.4.2

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

$t \geq 3 \sqrt{5}$

Step 2.4.3

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

$t < 0$

Step 2.4.4

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

$- t \geq 3 \sqrt{5}$

Step 2.4.5

Write as a piecewise.

${\begin{cases} t \geq 3 \sqrt{5} & t \geq 0 \\ - t \geq 3 \sqrt{5} & t < 0 \end{cases}$

Step 2.5

Find the intersection of $t \geq 3 \sqrt{5}$ and $t \geq 0$ .

$t \geq 3 \sqrt{5}$

Step 2.6

Divide each term in $- t \geq 3 \sqrt{5}$ by $- 1$ and simplify.

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Step 2.6.1

Divide each term in $- t \geq 3 \sqrt{5}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- t}{- 1} \leq \frac{3 \sqrt{5}}{- 1}$

Step 2.6.2

Simplify the left side.

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Step 2.6.2.1

Dividing two negative values results in a positive value.

$\frac{t}{1} \leq \frac{3 \sqrt{5}}{- 1}$

Step 2.6.2.2

Divide $t$ by $1$ .

$t \leq \frac{3 \sqrt{5}}{- 1}$

Step 2.6.3

Simplify the right side.

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Step 2.6.3.1

Move the negative one from the denominator of $\frac{3 \sqrt{5}}{- 1}$ .

$t \leq - 1 \cdot (3 \sqrt{5})$

Step 2.6.3.2

Rewrite $- 1 \cdot (3 \sqrt{5})$ as $- (3 \sqrt{5})$ .

$t \leq - (3 \sqrt{5})$

Step 2.6.3.3

Multiply $3$ by $- 1$ .

$t \leq - 3 \sqrt{5}$

Step 2.7

Find the union of the solutions.

$t \leq - 3 \sqrt{5}$ or $t \geq 3 \sqrt{5}$

Step 3

Set the denominator in $\frac{t}{\sqrt{t^{2} - 45}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{t^{2} - 45} = 0$

Step 4

Solve for $t$ .

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Step 4.1

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

${\sqrt{t^{2} - 45}}^{2} = 0^{2}$

Step 4.2

Simplify each side of the equation.

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Step 4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t^{2} - 45}$ as ${(t^{2} - 45)}^{\frac{1}{2}}$ .

${({(t^{2} - 45)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Simplify ${({(t^{2} - 45)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.1

Multiply the exponents in ${({(t^{2} - 45)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.1.1

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

${(t^{2} - 45)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.2.1.1.2.1

Cancel the common factor.

${(t^{2} - 45)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.1.2.2

Rewrite the expression.

${(t^{2} - 45)}^{1} = 0^{2}$

Step 4.2.2.1.2

Simplify.

$t^{2} - 45 = 0^{2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

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

$t^{2} - 45 = 0$

Step 4.3

Solve for $t$ .

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Step 4.3.1

Add $45$ to both sides of the equation.

$t^{2} = 45$

Step 4.3.2

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

$t = \pm \sqrt{45}$

Step 4.3.3

Simplify $\pm \sqrt{45}$ .

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Step 4.3.3.1

Rewrite $45$ as $3^{2} \cdot 5$ .

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Step 4.3.3.1.1

Factor $9$ out of $45$ .

$t = \pm \sqrt{9 (5)}$

Step 4.3.3.1.2

Rewrite $9$ as $3^{2}$ .

$t = \pm \sqrt{3^{2} \cdot 5}$

Step 4.3.3.2

Pull terms out from under the radical.

$t = \pm 3 \sqrt{5}$

Step 4.3.4

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

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Step 4.3.4.1

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

$t = 3 \sqrt{5}$

Step 4.3.4.2

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

$t = - 3 \sqrt{5}$

Step 4.3.4.3

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

$t = 3 \sqrt{5}, - 3 \sqrt{5}$

Step 4.4

Exclude the solutions that do not make $\sqrt{t^{2} - 45} = 0$ true.

$No solution$

Step 5

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

Interval Notation:

$(- \infty, - 3 \sqrt{5}] \cup [3 \sqrt{5}, \infty)$

Set-Builder Notation:

${t | t \leq - 3 \sqrt{5}, t \geq 3 \sqrt{5}}$

Step 6