Find the Domain (fog)(x) , f(x)=1/(x+3) , g(x)=2/x

$(f o g) (x)$ , $f (x) = \frac{1}{x + 3}$ , $g (x) = \frac{2}{x}$

Set up the composite result function.

$f (g (x))$

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

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

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

$x = 0$

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

$(\frac{2}{x}) + 3 = 0$

Solve for $x$ .

Subtract $3$ from both sides of the equation.

$\frac{2}{x} = - 3$

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.

$x, 1$

The LCM of one and any expression is the expression.

$x$

Multiply each term in $\frac{2}{x} = - 3$ by $x$ to eliminate the fractions.

Multiply each term in $\frac{2}{x} = - 3$ by $x$ .

$\frac{2}{x} x = - 3 x$

Simplify the left side.

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{2}{x} x = - 3 x$

Rewrite the expression.

$2 = - 3 x$

Solve the equation.

Rewrite the equation as $- 3 x = 2$ .

$- 3 x = 2$

Divide each term in $- 3 x = 2$ by $- 3$ and simplify.

Divide each term in $- 3 x = 2$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{2}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{2}{- 3}$

Divide $x$ by $1$ .

$x = \frac{2}{- 3}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{2}{3}$

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

Interval Notation:

$(- \infty, - \frac{2}{3}) \cup (- \frac{2}{3}, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0, - \frac{2}{3}}$

Find the Domain (fog)(x) , f(x)=1/(x+3) , g(x)=2/x

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