Find the Domain and Range f(x)=-2/(3(x-5))+4

$f (x) = - \frac{2}{3 (x - 5)} + 4$

Step 1

Set the denominator in $\frac{2}{3 (x - 5)}$ equal to $0$ to find where the expression is undefined.

$3 (x - 5) = 0$

Step 2

Solve for $x$ .

Divide each term in $3 (x - 5) = 0$ by $3$ and simplify.

Divide each term in $3 (x - 5) = 0$ by $3$ .

$\frac{3 (x - 5)}{3} = \frac{0}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 (x - 5)}{3} = \frac{0}{3}$

Divide $x - 5$ by $1$ .

$x - 5 = \frac{0}{3}$

Simplify the right side.

Divide $0$ by $3$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Step 3

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

Interval Notation:

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

Set-Builder Notation:

${x | x \neq 5}$

Step 4

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

Interval Notation:

$(- \infty, 4) \cup (4, \infty)$

Set-Builder Notation:

${y | y \neq 4}$

Step 5

Determine the domain and range.

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

Range: $(- \infty, 4) \cup (4, \infty), {y | y \neq 4}$

Step 6