Find the Domain and Range f(x)=(2x)/(3/4x+1/3)

$f (x) = \frac{2 x}{\frac{3}{4} x + \frac{1}{3}}$

Step 1

Set the denominator in $\frac{2 x}{\frac{3}{4} x + \frac{1}{3}}$ equal to $0$ to find where the expression is undefined.

$\frac{3}{4} x + \frac{1}{3} = 0$

Step 2

Solve for $x$ .

Combine $\frac{3}{4}$ and $x$ .

$\frac{3 x}{4} + \frac{1}{3} = 0$

Subtract $\frac{1}{3}$ from both sides of the equation.

$\frac{3 x}{4} = - \frac{1}{3}$

Multiply both sides of the equation by $\frac{4}{3}$ .

$\frac{4}{3} \cdot \frac{3 x}{4} = \frac{4}{3} (- \frac{1}{3})$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{4}{3} \cdot \frac{3 x}{4}$ .

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4}{3} \cdot \frac{3 x}{4} = \frac{4}{3} (- \frac{1}{3})$

Rewrite the expression.

$\frac{1}{3} (3 x) = \frac{4}{3} (- \frac{1}{3})$

Cancel the common factor of $3$ .

Factor $3$ out of $3 x$ .

$\frac{1}{3} (3 (x)) = \frac{4}{3} (- \frac{1}{3})$

Cancel the common factor.

$\frac{1}{3} (3 x) = \frac{4}{3} (- \frac{1}{3})$

Rewrite the expression.

$x = \frac{4}{3} (- \frac{1}{3})$

Simplify the right side.

Multiply $\frac{4}{3} (- \frac{1}{3})$ .

Multiply $\frac{4}{3}$ by $\frac{1}{3}$ .

$x = - \frac{4}{3 \cdot 3}$

Multiply $3$ by $3$ .

$x = - \frac{4}{9}$

Step 3

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

Interval Notation:

$(- \infty, - \frac{4}{9}) \cup (- \frac{4}{9}, \infty)$

Set-Builder Notation:

${x | x \neq - \frac{4}{9}}$

Step 4

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

Interval Notation:

$(- \infty, \frac{8}{3}) \cup (\frac{8}{3}, \infty)$

Set-Builder Notation:

${y | y \neq \frac{8}{3}}$

Step 5

Determine the domain and range.

Domain: $(- \infty, - \frac{4}{9}) \cup (- \frac{4}{9}, \infty), {x | x \neq - \frac{4}{9}}$

Range: $(- \infty, \frac{8}{3}) \cup (\frac{8}{3}, \infty), {y | y \neq \frac{8}{3}}$

Step 6

Find the Domain and Range f(x)=(2x)/(3/4x+1/3)

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