Algebra Examples

$f (x) = \frac{x^{2} + 5 x + 4}{3 x^{2} - 12}$

Step 1

Set the denominator in $\frac{x^{2} + 5 x + 4}{3 x^{2} - 12}$ equal to $0$ to find where the expression is undefined.

$3 x^{2} - 12 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Add $12$ to both sides of the equation.

$3 x^{2} = 12$

Step 2.2

Divide each term in $3 x^{2} = 12$ by $3$ and simplify.

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Step 2.2.1

Divide each term in $3 x^{2} = 12$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{12}{3}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $3$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{12}{3}$

Step 2.2.2.1.2

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

$x^{2} = \frac{12}{3}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $12$ by $3$ .

$x^{2} = 4$

Step 2.3

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

$x = \pm \sqrt{4}$

Step 2.4

Simplify $\pm \sqrt{4}$ .

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Step 2.4.1

Rewrite $4$ as $2^{2}$ .

$x = \pm \sqrt{2^{2}}$

Step 2.4.2

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

$x = \pm 2$

Step 2.5

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

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Step 2.5.1

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

$x = 2$

Step 2.5.2

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

$x = - 2$

Step 2.5.3

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

$x = 2, - 2$

Step 3

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

Interval Notation:

$(- \infty, - 2) \cup (- 2, 2) \cup (2, \infty)$

Set-Builder Notation:

${x | x \neq 2, - 2}$

Step 4

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

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${y | y \in ℝ}$

Step 5

Determine the domain and range.

Domain: $(- \infty, - 2) \cup (- 2, 2) \cup (2, \infty), {x | x \neq 2, - 2}$

Range: $(- \infty, \infty), {y | y \in ℝ}$

Step 6