Find the Domain and Range x/(6x-5)

$\frac{x}{6 x - 5}$

Step 1

Set the denominator in $\frac{x}{6 x - 5}$ equal to $0$ to find where the expression is undefined.

$6 x - 5 = 0$

Step 2

Solve for $x$ .

Add $5$ to both sides of the equation.

$6 x = 5$

Divide each term in $6 x = 5$ by $6$ and simplify.

Divide each term in $6 x = 5$ by $6$ .

$\frac{6 x}{6} = \frac{5}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} = \frac{5}{6}$

Divide $x$ by $1$ .

$x = \frac{5}{6}$

Step 3

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

Interval Notation:

$(- \infty, \frac{5}{6}) \cup (\frac{5}{6}, \infty)$

Set-Builder Notation:

${x | x \neq \frac{5}{6}}$

Step 4

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

Interval Notation:

$(- \infty, \frac{1}{6}) \cup (\frac{1}{6}, \infty)$

Set-Builder Notation:

${y | y \neq \frac{1}{6}}$

Step 5

Determine the domain and range.

Domain: $(- \infty, \frac{5}{6}) \cup (\frac{5}{6}, \infty), {x | x \neq \frac{5}{6}}$

Range: $(- \infty, \frac{1}{6}) \cup (\frac{1}{6}, \infty), {y | y \neq \frac{1}{6}}$

Step 6