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

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

Step 1

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

$2 x = 0$

Step 2

Divide each term in $2 x = 0$ by $2$ and simplify.

Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{0}{2}$

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

$x = 0$

Step 3

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

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Step 4

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

Interval Notation:

$(- \infty, \frac{1}{2}) \cup (\frac{1}{2}, \infty)$

Set-Builder Notation:

${y | y \neq \frac{1}{2}}$

Step 5

Determine the domain and range.

Domain: $(- \infty, 0) \cup (0, \infty), {x | x \neq 0}$

Range: $(- \infty, \frac{1}{2}) \cup (\frac{1}{2}, \infty), {y | y \neq \frac{1}{2}}$

Step 6