Calculus Examples

$f (x) = \frac{x}{x^{2} + x + 3}$

Step 1

Find the domain to determine if the expression is continuous.

Tap for more steps...

Step 1.1

Set the denominator in $\frac{x}{x^{2} + x + 3}$ equal to $0$ to find where the expression is undefined.

$x^{2} + x + 3 = 0$

Step 1.2

Solve for $x$ .

Tap for more steps...

Step 1.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 1.2.2

Substitute the values $a = 1$ , $b = 1$ , and $c = 3$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (1 \cdot 3)}}{2 \cdot 1}$

Step 1.2.3

Simplify.

Tap for more steps...

Step 1.2.3.1

Simplify the numerator.

Tap for more steps...

Step 1.2.3.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 3}}{2 \cdot 1}$

Step 1.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 3$ .

Tap for more steps...

Step 1.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 3}}{2 \cdot 1}$

Step 1.2.3.1.2.2

Multiply $- 4$ by $3$ .

$x = \frac{- 1 \pm \sqrt{1 - 12}}{2 \cdot 1}$

Step 1.2.3.1.3

Subtract $12$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 11}}{2 \cdot 1}$

Step 1.2.3.1.4

Rewrite $- 11$ as $- 1 (11)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 11}}{2 \cdot 1}$

Step 1.2.3.1.5

Rewrite $\sqrt{- 1 (11)}$ as $\sqrt{- 1} \cdot \sqrt{11}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{11}}{2 \cdot 1}$

Step 1.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \sqrt{11}}{2 \cdot 1}$

Step 1.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm i \sqrt{11}}{2}$

Step 1.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Tap for more steps...

Step 1.2.4.1

Simplify the numerator.

Tap for more steps...

Step 1.2.4.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 3}}{2 \cdot 1}$

Step 1.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 3$ .

Tap for more steps...

Step 1.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 3}}{2 \cdot 1}$

Step 1.2.4.1.2.2

Multiply $- 4$ by $3$ .

$x = \frac{- 1 \pm \sqrt{1 - 12}}{2 \cdot 1}$

Step 1.2.4.1.3

Subtract $12$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 11}}{2 \cdot 1}$

Step 1.2.4.1.4

Rewrite $- 11$ as $- 1 (11)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 11}}{2 \cdot 1}$

Step 1.2.4.1.5

Rewrite $\sqrt{- 1 (11)}$ as $\sqrt{- 1} \cdot \sqrt{11}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{11}}{2 \cdot 1}$

Step 1.2.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \sqrt{11}}{2 \cdot 1}$

Step 1.2.4.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm i \sqrt{11}}{2}$

Step 1.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{- 1 + i \sqrt{11}}{2}$

Step 1.2.4.4

Rewrite $- 1$ as $- 1 (1)$ .

$x = \frac{- 1 \cdot 1 + i \sqrt{11}}{2}$

Step 1.2.4.5

Factor $- 1$ out of $i \sqrt{11}$ .

$x = \frac{- 1 \cdot 1 - (- i \sqrt{11})}{2}$

Step 1.2.4.6

Factor $- 1$ out of $- 1 (1) - (- i \sqrt{11})$ .

$x = \frac{- 1 (1 - i \sqrt{11})}{2}$

Step 1.2.4.7

Move the negative in front of the fraction.

$x = - \frac{1 - i \sqrt{11}}{2}$

Step 1.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Tap for more steps...

Step 1.2.5.1

Simplify the numerator.

Tap for more steps...

Step 1.2.5.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 3}}{2 \cdot 1}$

Step 1.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 3$ .

Tap for more steps...

Step 1.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 3}}{2 \cdot 1}$

Step 1.2.5.1.2.2

Multiply $- 4$ by $3$ .

$x = \frac{- 1 \pm \sqrt{1 - 12}}{2 \cdot 1}$

Step 1.2.5.1.3

Subtract $12$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 11}}{2 \cdot 1}$

Step 1.2.5.1.4

Rewrite $- 11$ as $- 1 (11)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 11}}{2 \cdot 1}$

Step 1.2.5.1.5

Rewrite $\sqrt{- 1 (11)}$ as $\sqrt{- 1} \cdot \sqrt{11}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{11}}{2 \cdot 1}$

Step 1.2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \sqrt{11}}{2 \cdot 1}$

Step 1.2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm i \sqrt{11}}{2}$

Step 1.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{- 1 - i \sqrt{11}}{2}$

Step 1.2.5.4

Rewrite $- 1$ as $- 1 (1)$ .

$x = \frac{- 1 \cdot 1 - i \sqrt{11}}{2}$

Step 1.2.5.5

Factor $- 1$ out of $- i \sqrt{11}$ .

$x = \frac{- 1 \cdot 1 - (i \sqrt{11})}{2}$

Step 1.2.5.6

Factor $- 1$ out of $- 1 (1) - (i \sqrt{11})$ .

$x = \frac{- 1 (1 + i \sqrt{11})}{2}$

Step 1.2.5.7

Move the negative in front of the fraction.

$x = - \frac{1 + i \sqrt{11}}{2}$

Step 1.2.6

The final answer is the combination of both solutions.

$x = - \frac{1 - i \sqrt{11}}{2}, - \frac{1 + i \sqrt{11}}{2}$

Step 1.3

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 2

Since the domain is all real numbers, $\frac{x}{x^{2} + x + 3}$ is continuous over all real numbers.

Continuous

Step 3