Find the Domain and Range y=(12x-5)/(3x-1)

$y = \frac{12 x - 5}{3 x - 1}$

Step 1

Set the denominator in $\frac{12 x - 5}{3 x - 1}$ equal to $0$ to find where the expression is undefined.

$3 x - 1 = 0$

Step 2

Solve for $x$ .

Add $1$ to both sides of the equation.

$3 x = 1$

Divide each term in $3 x = 1$ by $3$ and simplify.

Divide each term in $3 x = 1$ by $3$ .

$\frac{3 x}{3} = \frac{1}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{1}{3}$

Divide $x$ by $1$ .

$x = \frac{1}{3}$

Step 3

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

Interval Notation:

$(- \infty, \frac{1}{3}) \cup (\frac{1}{3}, \infty)$

Set-Builder Notation:

${x | x \neq \frac{1}{3}}$

Step 4

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

Interval Notation:

$(- \infty, 4) \cup (4, \infty)$

Set-Builder Notation:

${y | y \neq 4}$

Step 5

Determine the domain and range.

Domain: $(- \infty, \frac{1}{3}) \cup (\frac{1}{3}, \infty), {x | x \neq \frac{1}{3}}$

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

Step 6