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

$f (x) = \frac{x}{x^{2} - 2 x}$

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

$x^{2} - 2 x = 0$

Solve for $x$ .

Factor $x$ out of $x^{2} - 2 x$ .

Factor $x$ out of $x^{2}$ .

$x \cdot x - 2 x = 0$

Factor $x$ out of $- 2 x$ .

$x \cdot x + x \cdot - 2 = 0$

Factor $x$ out of $x \cdot x + x \cdot - 2$ .

$x (x - 2) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 2 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

The final solution is all the values that make $x (x - 2) = 0$ true.

$x = 0, 2$

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

Interval Notation:

$(- \infty, 0) \cup (0, 2) \cup (2, \infty)$

Set-Builder Notation:

${x | x \neq 0, 2}$

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

Interval Notation:

$(- \infty, - \frac{1}{2}) \cup (- \frac{1}{2}, 0) \cup (0, \infty)$

Set-Builder Notation:

${y | y \neq 0, - \frac{1}{2}}$

Determine the domain and range.

Domain: $(- \infty, 0) \cup (0, 2) \cup (2, \infty), {x | x \neq 0, 2}$

Range: $(- \infty, - \frac{1}{2}) \cup (- \frac{1}{2}, 0) \cup (0, \infty), {y | y \neq 0, - \frac{1}{2}}$