Finite Math Examples

$y = \sqrt{t^{2} - 2 t} - 3$

Step 1

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

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

Step 2

Solve for $t$ .

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Step 2.1

Convert the inequality to an equation.

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

Step 2.2

Factor $t$ out of $t^{2} - 2 t$ .

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Step 2.2.1

Factor $t$ out of $t^{2}$ .

$t \cdot t - 2 t = 0$

Step 2.2.2

Factor $t$ out of $- 2 t$ .

$t \cdot t + t \cdot - 2 = 0$

Step 2.2.3

Factor $t$ out of $t \cdot t + t \cdot - 2$ .

$t (t - 2) = 0$

Step 2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$t = 0$

$t - 2 = 0$

Step 2.4

Set $t$ equal to $0$ .

$t = 0$

Step 2.5

Set $t - 2$ equal to $0$ and solve for $t$ .

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Step 2.5.1

Set $t - 2$ equal to $0$ .

$t - 2 = 0$

Step 2.5.2

Add $2$ to both sides of the equation.

$t = 2$

Step 2.6

The final solution is all the values that make $t (t - 2) = 0$ true.

$t = 0, 2$

Step 2.7

Use each root to create test intervals.

$t < 0$

$0 < t < 2$

$t > 2$

Step 2.8

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 2.8.1

Test a value on the interval $t < 0$ to see if it makes the inequality true.

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Step 2.8.1.1

Choose a value on the interval $t < 0$ and see if this value makes the original inequality true.

$t = - 2$

Step 2.8.1.2

Replace $t$ with $- 2$ in the original inequality.

${(- 2)}^{2} - 2 \cdot - 2 \geq 0$

Step 2.8.1.3

The left side $8$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 2.8.2

Test a value on the interval $0 < t < 2$ to see if it makes the inequality true.

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Step 2.8.2.1

Choose a value on the interval $0 < t < 2$ and see if this value makes the original inequality true.

$t = 1$

Step 2.8.2.2

Replace $t$ with $1$ in the original inequality.

${(1)}^{2} - 2 \cdot 1 \geq 0$

Step 2.8.2.3

The left side $- 1$ is less than the right side $0$ , which means that the given statement is false.

False

Step 2.8.3

Test a value on the interval $t > 2$ to see if it makes the inequality true.

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Step 2.8.3.1

Choose a value on the interval $t > 2$ and see if this value makes the original inequality true.

$t = 4$

Step 2.8.3.2

Replace $t$ with $4$ in the original inequality.

${(4)}^{2} - 2 \cdot 4 \geq 0$

Step 2.8.3.3

The left side $8$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 2.8.4

Compare the intervals to determine which ones satisfy the original inequality.

$t < 0$ True

$0 < t < 2$ False

$t > 2$ True

$t < 0$ True

$0 < t < 2$ False

$t > 2$ True

Step 2.9

The solution consists of all of the true intervals.

$t \leq 0$ or $t \geq 2$

Step 3

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

Interval Notation:

$(- \infty, 0] \cup [2, \infty)$

Set-Builder Notation:

${t | t \leq 0, t \geq 2}$

Step 4