Precalculus Examples

$R (x) = \frac{4 x + 4}{7 x + 21}$

Step 1

Set the denominator in $\frac{4 x + 4}{7 x + 21}$ equal to $0$ to find where the expression is undefined.

$7 x + 21 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $21$ from both sides of the equation.

$7 x = - 21$

Step 2.2

Divide each term in $7 x = - 21$ by $7$ and simplify.

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Step 2.2.1

Divide each term in $7 x = - 21$ by $7$ .

$\frac{7 x}{7} = \frac{- 21}{7}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $7$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{- 21}{7}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 21}{7}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $- 21$ by $7$ .

$x = - 3$

Step 3

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

Interval Notation:

$(- \infty, - 3) \cup (- 3, \infty)$

Set-Builder Notation:

${x | x \neq - 3}$

Step 4

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

Interval Notation:

$(- \infty, \frac{4}{7}) \cup (\frac{4}{7}, \infty)$

Set-Builder Notation:

${y | y \neq \frac{4}{7}}$

Step 5

Determine the domain and range.

Domain: $(- \infty, - 3) \cup (- 3, \infty), {x | x \neq - 3}$

Range: $(- \infty, \frac{4}{7}) \cup (\frac{4}{7}, \infty), {y | y \neq \frac{4}{7}}$

Step 6