Calculus Examples

$f (x) = \frac{x^{2} + 3 x + 4}{\sqrt{x^{2} + 1}}$

Step 1

Set the radicand in $\sqrt{x^{2} + 1}$ greater than or equal to $0$ to find where the expression is defined.

$x^{2} + 1 \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $1$ from both sides of the inequality.

$x^{2} \geq - 1$

Step 2.2

Since the left side has an even power, it is always positive for all real numbers.

All real numbers

Step 3

Set the denominator in $\frac{x^{2} + 3 x + 4}{\sqrt{x^{2} + 1}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{x^{2} + 1} = 0$

Step 4

Solve for $x$ .

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Step 4.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x^{2} + 1}}^{2} = 0^{2}$

Step 4.2

Simplify each side of the equation.

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Step 4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 1}$ as ${(x^{2} + 1)}^{\frac{1}{2}}$ .

${({(x^{2} + 1)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Simplify ${({(x^{2} + 1)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.1

Multiply the exponents in ${({(x^{2} + 1)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x^{2} + 1)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.2.1.1.2.1

Cancel the common factor.

${(x^{2} + 1)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.1.2.2

Rewrite the expression.

${(x^{2} + 1)}^{1} = 0^{2}$

Step 4.2.2.1.2

Simplify.

$x^{2} + 1 = 0^{2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Raising $0$ to any positive power yields $0$ .

$x^{2} + 1 = 0$

Step 4.3

Solve for $x$ .

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Step 4.3.1

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Step 4.3.2

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

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

Step 4.3.3

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

$x = \pm i$

Step 4.3.4

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

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Step 4.3.4.1

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

$x = i$

Step 4.3.4.2

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

$x = - i$

Step 4.3.4.3

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

$x = i, - i$

Step 5

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 6