Calculus Examples

$f (x) = \sqrt{2 - 3 x}$

Step 1

Set the radicand in $\sqrt{2 - 3 x}$ greater than or equal to $0$ to find where the expression is defined.

$2 - 3 x \geq 0$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

Subtract $2$ from both sides of the inequality.

$- 3 x \geq - 2$

Step 2.2

Divide each term in $- 3 x \geq - 2$ by $- 3$ and simplify.

Tap for more steps...

Step 2.2.1

Divide each term in $- 3 x \geq - 2$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 x}{- 3} \leq \frac{- 2}{- 3}$

Step 2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.1

Cancel the common factor of $- 3$ .

Tap for more steps...

Step 2.2.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} \leq \frac{- 2}{- 3}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 2}{- 3}$

Step 2.2.3

Simplify the right side.

Tap for more steps...

Step 2.2.3.1

Dividing two negative values results in a positive value.

$x \leq \frac{2}{3}$

Step 3

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

Interval Notation:

$(- \infty, \frac{2}{3}]$

Set-Builder Notation:

${x | x \leq \frac{2}{3}}$

Step 4

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${y | y \geq 0}$

Step 5

Determine the domain and range.

Domain: $(- \infty, \frac{2}{3}], {x | x \leq \frac{2}{3}}$

Range: $[0, \infty), {y | y \geq 0}$

Step 6