Calculus Examples

$f (x) = \ln (11 x^{2} + 6)$

Step 1

Set the argument in $\ln (11 x^{2} + 6)$ greater than $0$ to find where the expression is defined.

$11 x^{2} + 6 > 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $6$ from both sides of the inequality.

$11 x^{2} > - 6$

Step 2.2

Divide each term in $11 x^{2} > - 6$ by $11$ and simplify.

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Step 2.2.1

Divide each term in $11 x^{2} > - 6$ by $11$ .

$\frac{11 x^{2}}{11} > \frac{- 6}{11}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $11$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{11 x^{2}}{11} > \frac{- 6}{11}$

Step 2.2.2.1.2

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

$x^{2} > \frac{- 6}{11}$

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.

$x^{2} > - \frac{6}{11}$

Step 2.3

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

All real numbers

Step 3

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4

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

Interval Notation:

$[\ln (6), \infty)$

Set-Builder Notation:

${y | y \geq \ln (6)}$

Step 5

Determine the domain and range.

Domain: $(- \infty, \infty), {x | x \in ℝ}$

Range: $[\ln (6), \infty), {y | y \geq \ln (6)}$

Step 6