Find the Domain f(x)=(4x)/(9-x)

$f (x) = \frac{4 x}{9 - x}$

Step 1

Set the denominator in $\frac{4 x}{9 - x}$ equal to $0$ to find where the expression is undefined.

$9 - x = 0$

Step 2

Solve for $x$ .

Subtract $9$ from both sides of the equation.

$- x = - 9$

Divide each term in $- x = - 9$ by $- 1$ and simplify.

Divide each term in $- x = - 9$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 9}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 9}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 9}{- 1}$

Simplify the right side.

Divide $- 9$ by $- 1$ .

$x = 9$

Step 3

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

Interval Notation:

$(- \infty, 9) \cup (9, \infty)$

Set-Builder Notation:

${x | x \neq 9}$

Step 4