Precalculus Examples

$\frac{2 x + 8}{- 8 x + 4}$

Step 1

Set the denominator in $\frac{2 x + 8}{- 8 x + 4}$ equal to $0$ to find where the expression is undefined.

$- 8 x + 4 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $4$ from both sides of the equation.

$- 8 x = - 4$

Step 2.2

Divide each term in $- 8 x = - 4$ by $- 8$ and simplify.

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Step 2.2.1

Divide each term in $- 8 x = - 4$ by $- 8$ .

$\frac{- 8 x}{- 8} = \frac{- 4}{- 8}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $- 8$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{- 8 x}{- 8} = \frac{- 4}{- 8}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 8}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Cancel the common factor of $- 4$ and $- 8$ .

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Step 2.2.3.1.1

Factor $- 4$ out of $- 4$ .

$x = \frac{- 4 (1)}{- 8}$

Step 2.2.3.1.2

Cancel the common factors.

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Step 2.2.3.1.2.1

Factor $- 4$ out of $- 8$ .

$x = \frac{- 4 \cdot 1}{- 4 \cdot 2}$

Step 2.2.3.1.2.2

Cancel the common factor.

$x = \frac{- 4 \cdot 1}{- 4 \cdot 2}$

Step 2.2.3.1.2.3

Rewrite the expression.

$x = \frac{1}{2}$

Step 3

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

Interval Notation:

$(- \infty, \frac{1}{2}) \cup (\frac{1}{2}, \infty)$

Set-Builder Notation:

${x | x \neq \frac{1}{2}}$

Step 4

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

Interval Notation:

$(- \infty, - \frac{1}{4}) \cup (- \frac{1}{4}, \infty)$

Set-Builder Notation:

${y | y \neq - \frac{1}{4}}$

Step 5

Determine the domain and range.

Domain: $(- \infty, \frac{1}{2}) \cup (\frac{1}{2}, \infty), {x | x \neq \frac{1}{2}}$

Range: $(- \infty, - \frac{1}{4}) \cup (- \frac{1}{4}, \infty), {y | y \neq - \frac{1}{4}}$

Step 6