Find the Domain and Range f(x)=-2/(2x-4)+1

$f (x) = - \frac{2}{2 x - 4} + 1$

Step 1

Set the denominator in $\frac{2}{2 x - 4}$ equal to $0$ to find where the expression is undefined.

$2 x - 4 = 0$

Step 2

Solve for $x$ .

Add $4$ to both sides of the equation.

$2 x = 4$

Divide each term in $2 x = 4$ by $2$ and simplify.

Divide each term in $2 x = 4$ by $2$ .

$\frac{2 x}{2} = \frac{4}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{4}{2}$

Divide $x$ by $1$ .

$x = \frac{4}{2}$

Simplify the right side.

Divide $4$ by $2$ .

$x = 2$

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:

$(- \infty, 1) \cup (1, \infty)$

Set-Builder Notation:

${y | y \neq 1}$

Step 5

Determine the domain and range.

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

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

Step 6