Calculus Examples

$f (x) = \frac{4 x}{x^{2} + 49}$

Step 1

Find the domain to determine if the expression is continuous.

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Step 1.1

Set the denominator in $\frac{4 x}{x^{2} + 49}$ equal to $0$ to find where the expression is undefined.

$x^{2} + 49 = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Subtract $49$ from both sides of the equation.

$x^{2} = - 49$

Step 1.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 49}$

Step 1.2.3

Simplify $\pm \sqrt{- 49}$ .

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Step 1.2.3.1

Rewrite $- 49$ as $- 1 (49)$ .

$x = \pm \sqrt{- 1 (49)}$

Step 1.2.3.2

Rewrite $\sqrt{- 1 (49)}$ as $\sqrt{- 1} \cdot \sqrt{49}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{49}$

Step 1.2.3.3

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

$x = \pm i \cdot \sqrt{49}$

Step 1.2.3.4

Rewrite $49$ as $7^{2}$ .

$x = \pm i \cdot \sqrt{7^{2}}$

Step 1.2.3.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm i \cdot 7$

Step 1.2.3.6

Move $7$ to the left of $i$ .

$x = \pm 7 i$

Step 1.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 7 i$

Step 1.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 7 i$

Step 1.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 7 i, - 7 i$

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, $\frac{4 x}{x^{2} + 49}$ is continuous over all real numbers.

Continuous

Step 3