Find the Domain and Range f(x)=2/(x(x-5))

$f (x) = \frac{2}{x (x - 5)}$

Step 1

Set the denominator in $\frac{2}{x (x - 5)}$ equal to $0$ to find where the expression is undefined.

$x (x - 5) = 0$

Step 2

Solve for $x$ .

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

$x = 0$

$x - 5 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

The final solution is all the values that make $x (x - 5) = 0$ true.

$x = 0, 5$

Step 3

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

Interval Notation:

$(- \infty, 0) \cup (0, 5) \cup (5, \infty)$

Set-Builder Notation:

${x | x \neq 0, 5}$

Step 4

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

Interval Notation:

$(- \infty, - \frac{8}{25}] \cup (0, \infty)$

Set-Builder Notation:

${y | y \leq - \frac{8}{25}, y > 0}$

Step 5

Determine the domain and range.

Domain: $(- \infty, 0) \cup (0, 5) \cup (5, \infty), {x | x \neq 0, 5}$

Range: $(- \infty, - \frac{8}{25}] \cup (0, \infty), {y | y \leq - \frac{8}{25}, y > 0}$

Step 6