Finite Math Examples

$\frac{t - 3}{2 - \sqrt{t}}$

Step 1

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

$t \geq 0$

Step 2

Set the denominator in $\frac{t - 3}{2 - \sqrt{t}}$ equal to $0$ to find where the expression is undefined.

$2 - \sqrt{t} = 0$

Step 3

Solve for $t$ .

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Step 3.1

Subtract $2$ from both sides of the equation.

$- \sqrt{t} = - 2$

Step 3.2

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

${(- \sqrt{t})}^{2} = {(- 2)}^{2}$

Step 3.3

Simplify each side of the equation.

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Step 3.3.1

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

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

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

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Step 3.3.2.1.1

Apply the product rule to $- t^{\frac{1}{2}}$ .

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

Step 3.3.2.1.2

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

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

Step 3.3.2.1.3

Multiply ${(t^{\frac{1}{2}})}^{2}$ by $1$ .

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

Step 3.3.2.1.4

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

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Step 3.3.2.1.4.1

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

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

Step 3.3.2.1.4.2

Cancel the common factor of $2$ .

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Step 3.3.2.1.4.2.1

Cancel the common factor.

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

Step 3.3.2.1.4.2.2

Rewrite the expression.

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

Step 3.3.2.1.5

Simplify.

$t = {(- 2)}^{2}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

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

$t = 4$

Step 4

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

Interval Notation:

$[0, 4) \cup (4, \infty)$

Set-Builder Notation:

${t | t \geq 0, t \neq 4}$

Step 5