Precalculus Examples

$f (x) = \frac{x - 7}{x - 12 x + 35}$

Step 1

Set the denominator in $\frac{x - 7}{x - 12 x + 35}$ equal to $0$ to find where the expression is undefined.

$x - 12 x + 35 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $12 x$ from $x$ .

$- 11 x + 35 = 0$

Step 2.2

Subtract $35$ from both sides of the equation.

$- 11 x = - 35$

Step 2.3

Divide each term in $- 11 x = - 35$ by $- 11$ and simplify.

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Step 2.3.1

Divide each term in $- 11 x = - 35$ by $- 11$ .

$\frac{- 11 x}{- 11} = \frac{- 35}{- 11}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $- 11$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{- 11 x}{- 11} = \frac{- 35}{- 11}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 35}{- 11}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Dividing two negative values results in a positive value.

$x = \frac{35}{11}$

Step 3

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

Interval Notation:

$(- \infty, \frac{35}{11}) \cup (\frac{35}{11}, \infty)$

Set-Builder Notation:

${x | x \neq \frac{35}{11}}$

Step 4

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

Interval Notation:

$(- \infty, - \frac{1}{11}) \cup (- \frac{1}{11}, \infty)$

Set-Builder Notation:

${y | y \neq - \frac{1}{11}}$

Step 5

Determine the domain and range.

Domain: $(- \infty, \frac{35}{11}) \cup (\frac{35}{11}, \infty), {x | x \neq \frac{35}{11}}$

Range: $(- \infty, - \frac{1}{11}) \cup (- \frac{1}{11}, \infty), {y | y \neq - \frac{1}{11}}$

Step 6