Find the Domain and Range y=(2x-1)/(2-x)

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

Step 1

Set the denominator in $\frac{2 x - 1}{2 - x}$ equal to $0$ to find where the expression is undefined.

$2 - x = 0$

Step 2

Solve for $x$ .

Subtract $2$ from both sides of the equation.

$- x = - 2$

Divide each term in $- x = - 2$ by $- 1$ and simplify.

Divide each term in $- x = - 2$ by $- 1$ .

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $- 2$ by $- 1$ .

$x = 2$

Step 3

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

Interval Notation:

$(- \infty, 2) \cup (2, \infty)$

Set-Builder Notation:

${x | x \neq 2}$

Step 4

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

Interval Notation:

$(- \infty, - 2) \cup (- 2, \infty)$

Set-Builder Notation:

${y | y \neq - 2}$

Step 5

Determine the domain and range.

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

Range: $(- \infty, - 2) \cup (- 2, \infty), {y | y \neq - 2}$

Step 6