Precalculus Examples

$P (t) = \frac{\sqrt{t - 4}}{3 t - 21}$

Step 1

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

$t - 4 \geq 0$

Step 2

Add $4$ to both sides of the inequality.

$t \geq 4$

Step 3

Set the denominator in $\frac{\sqrt{t - 4}}{3 t - 21}$ equal to $0$ to find where the expression is undefined.

$3 t - 21 = 0$

Step 4

Solve for $t$ .

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Step 4.1

Add $21$ to both sides of the equation.

$3 t = 21$

Step 4.2

Divide each term in $3 t = 21$ by $3$ and simplify.

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Step 4.2.1

Divide each term in $3 t = 21$ by $3$ .

$\frac{3 t}{3} = \frac{21}{3}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $3$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{3 t}{3} = \frac{21}{3}$

Step 4.2.2.1.2

Divide $t$ by $1$ .

$t = \frac{21}{3}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $21$ by $3$ .

$t = 7$

Step 5

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

Interval Notation:

$[4, 7) \cup (7, \infty)$

Set-Builder Notation:

${t | t \geq 4, t \neq 7}$

Step 6