Find the Domain f(x)=(4x^2)/(3x)+4 , g(x)=9/x , g(f(x))

$f (x) = \frac{4 x^{2}}{3 x} + 4$ , $g (x) = \frac{9}{x}$ , $g (f (x))$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (\frac{4 x^{2}}{3 x} + 4)$ by substituting in the value of $f$ into $g$ .

$g (\frac{4 x^{2}}{3 x} + 4) = \frac{9}{\frac{4 x^{2}}{3 x} + 4}$

Step 3

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

$3 x = 0$

Step 4

Divide each term in $3 x = 0$ by $3$ and simplify.

Divide each term in $3 x = 0$ by $3$ .

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $0$ by $3$ .

$x = 0$

Step 5

Set the denominator in $\frac{9}{\frac{4 x^{2}}{3 x} + 4}$ equal to $0$ to find where the expression is undefined.

$\frac{4 x^{2}}{3 x} + 4 = 0$

Step 6

Solve for $x$ .

Reduce the expression $\frac{4 x^{2}}{3 x}$ by cancelling the common factors.

Factor $x$ out of $4 x^{2}$ .

$\frac{x (4 x)}{3 x} + 4 = 0$

Factor $x$ out of $3 x$ .

$\frac{x (4 x)}{x \cdot 3} + 4 = 0$

Cancel the common factor.

$\frac{x (4 x)}{x \cdot 3} + 4 = 0$

Rewrite the expression.

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

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$3, 1, 1$

The LCM of one and any expression is the expression.

$3$

Multiply each term in $\frac{4 x}{3} + 4 = 0$ by $3$ to eliminate the fractions.

Multiply each term in $\frac{4 x}{3} + 4 = 0$ by $3$ .

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

Simplify the left side.

Simplify each term.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

$4 x + 4 \cdot 3 = 0 \cdot 3$

Multiply $4$ by $3$ .

$4 x + 12 = 0 \cdot 3$

Simplify the right side.

Multiply $0$ by $3$ .

$4 x + 12 = 0$

Solve the equation.

Subtract $12$ from both sides of the equation.

$4 x = - 12$

Divide each term in $4 x = - 12$ by $4$ and simplify.

Divide each term in $4 x = - 12$ by $4$ .

$\frac{4 x}{4} = \frac{- 12}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 12}{4}$

Divide $x$ by $1$ .

$x = \frac{- 12}{4}$

Simplify the right side.

Divide $- 12$ by $4$ .

$x = - 3$

Step 7

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

Interval Notation:

$(- \infty, - 3) \cup (- 3, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0, - 3}$

Step 8

Find the Domain f(x)=(4x^2)/(3x)+4 , g(x)=9/x , g(f(x))

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