Calculus Examples

$f (x) = \frac{\sqrt{36 x^{2} + 7}}{9 x + 4}$

Step 1

Set the radicand in $\sqrt{36 x^{2} + 7}$ greater than or equal to $0$ to find where the expression is defined.

$36 x^{2} + 7 \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $7$ from both sides of the inequality.

$36 x^{2} \geq - 7$

Step 2.2

Divide each term in $36 x^{2} \geq - 7$ by $36$ and simplify.

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Step 2.2.1

Divide each term in $36 x^{2} \geq - 7$ by $36$ .

$\frac{36 x^{2}}{36} \geq \frac{- 7}{36}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $36$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{36 x^{2}}{36} \geq \frac{- 7}{36}$

Step 2.2.2.1.2

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

$x^{2} \geq \frac{- 7}{36}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Move the negative in front of the fraction.

$x^{2} \geq - \frac{7}{36}$

Step 2.3

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 $\frac{\sqrt{36 x^{2} + 7}}{9 x + 4}$ equal to $0$ to find where the expression is undefined.

$9 x + 4 = 0$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $4$ from both sides of the equation.

$9 x = - 4$

Step 4.2

Divide each term in $9 x = - 4$ by $9$ and simplify.

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Step 4.2.1

Divide each term in $9 x = - 4$ by $9$ .

$\frac{9 x}{9} = \frac{- 4}{9}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $9$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{- 4}{9}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{9}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Move the negative in front of the fraction.

$x = - \frac{4}{9}$

Step 5

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

Interval Notation:

$(- \infty, - \frac{4}{9}) \cup (- \frac{4}{9}, \infty)$

Set-Builder Notation:

${x | x \neq - \frac{4}{9}}$

Step 6

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

Interval Notation:

$(- \infty, - \frac{2}{3}) \cup [\frac{2 \sqrt{889}}{127}, \infty)$

Set-Builder Notation:

${y | y < - \frac{2}{3}, y \geq \frac{2 \sqrt{889}}{127}}$

Step 7

Determine the domain and range.

Domain: $(- \infty, - \frac{4}{9}) \cup (- \frac{4}{9}, \infty), {x | x \neq - \frac{4}{9}}$

Range: $(- \infty, - \frac{2}{3}) \cup [\frac{2 \sqrt{889}}{127}, \infty), {y | y < - \frac{2}{3}, y \geq \frac{2 \sqrt{889}}{127}}$

Step 8