Calculus Examples

$g (t) = \sqrt{3 - t} - \sqrt{2 + t}$

Step 1

Set the radicand in $\sqrt{3 - t}$ greater than or equal to $0$ to find where the expression is defined.

$3 - t \geq 0$

Step 2

Solve for $t$ .

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Step 2.1

Subtract $3$ from both sides of the inequality.

$- t \geq - 3$

Step 2.2

Divide each term in $- t \geq - 3$ by $- 1$ and simplify.

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Step 2.2.1

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

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

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.

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

Step 2.2.2.2

Divide $t$ by $1$ .

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

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $- 3$ by $- 1$ .

$t \leq 3$

Step 3

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

$2 + t \geq 0$

Step 4

Subtract $2$ from both sides of the inequality.

$t \geq - 2$

Step 5

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

Interval Notation:

$[- 2, 3]$

Set-Builder Notation:

${t | - 2 \leq t \leq 3}$

Step 6