Precalculus Examples

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

Step 1

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

$3 - e^{2 x} \geq 0$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

Subtract $3$ from both sides of the inequality.

$- e^{2 x} \geq - 3$

Step 2.2

Divide each term in $- e^{2 x} \geq - 3$ by $- 1$ and simplify.

Tap for more steps...

Step 2.2.1

Divide each term in $- e^{2 x} \geq - 3$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- e^{2 x}}{- 1} \leq \frac{- 3}{- 1}$

Step 2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.1

Dividing two negative values results in a positive value.

$\frac{e^{2 x}}{1} \leq \frac{- 3}{- 1}$

Step 2.2.2.2

Divide $e^{2 x}$ by $1$ .

$e^{2 x} \leq \frac{- 3}{- 1}$

Step 2.2.3

Simplify the right side.

Tap for more steps...

Step 2.2.3.1

Divide $- 3$ by $- 1$ .

$e^{2 x} \leq 3$

Step 2.3

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{2 x}) \leq \ln (3)$

Step 2.4

Expand the left side.

Tap for more steps...

Step 2.4.1

Expand $\ln (e^{2 x})$ by moving $2 x$ outside the logarithm.

$2 x \ln (e) \leq \ln (3)$

Step 2.4.2

The natural logarithm of $e$ is $1$ .

$2 x \cdot 1 \leq \ln (3)$

Step 2.4.3

Multiply $2$ by $1$ .

$2 x \leq \ln (3)$

Step 2.5

Divide each term in $2 x \leq \ln (3)$ by $2$ and simplify.

Tap for more steps...

Step 2.5.1

Divide each term in $2 x \leq \ln (3)$ by $2$ .

$\frac{2 x}{2} \leq \frac{\ln (3)}{2}$

Step 2.5.2

Simplify the left side.

Tap for more steps...

Step 2.5.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.5.2.1.1

Cancel the common factor.

$\frac{2 x}{2} \leq \frac{\ln (3)}{2}$

Step 2.5.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{\ln (3)}{2}$

Step 3

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

Interval Notation:

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

Set-Builder Notation:

${x | x \leq \frac{\ln (3)}{2}}$

Step 4