Algebra Examples

(f o g) (x)

$(f o g) (x)$ ,

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

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

g (x) = \frac{2}{x}

$g (x) = \frac{2}{x}$

Step 1

Set up the composite result function.

f (g (x))

$f (g (x))$

Step 2

Evaluate

f (\frac{2}{x})

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

g

$g$ into

f

$f$ .

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

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

Step 3

Set the denominator in

\frac{2}{x}

$\frac{2}{x}$ equal to

0

$0$ to find where the expression is undefined.

x = 0

$x = 0$

Step 4

Set the denominator in

\frac{1}{(\frac{2}{x}) + 3}

$\frac{1}{(\frac{2}{x}) + 3}$ equal to

0

$0$ to find where the expression is undefined.

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

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

Step 5

Solve for

x

$x$ .

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Step 5.1

Subtract

3

$3$ from both sides of the equation.

\frac{2}{x} = - 3

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

Step 5.2

Find the LCD of the terms in the equation.

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Step 5.2.1

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

x, 1

$x, 1$

Step 5.2.2

The LCM of one and any expression is the expression.

x

$x$

x

$x$

Step 5.3

Multiply each term in

\frac{2}{x} = - 3

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

x

$x$ to eliminate the fractions.

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Step 5.3.1

Multiply each term in

\frac{2}{x} = - 3

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

x

$x$ .

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

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of

x

$x$ .

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Step 5.3.2.1.1

Cancel the common factor.

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

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

Step 5.3.2.1.2

Rewrite the expression.

2 = - 3 x

$2 = - 3 x$

2 = - 3 x

$2 = - 3 x$

2 = - 3 x

$2 = - 3 x$

2 = - 3 x

$2 = - 3 x$

Step 5.4

Solve the equation.

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Step 5.4.1

Rewrite the equation as

- 3 x = 2

$- 3 x = 2$ .

- 3 x = 2

$- 3 x = 2$

Step 5.4.2

Divide each term in

- 3 x = 2

$- 3 x = 2$ by

- 3

$- 3$ and simplify.

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Step 5.4.2.1

Divide each term in

- 3 x = 2

$- 3 x = 2$ by

- 3

$- 3$ .

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

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

Step 5.4.2.2

Simplify the left side.

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Step 5.4.2.2.1

Cancel the common factor of

- 3

$- 3$ .

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Step 5.4.2.2.1.1

Cancel the common factor.

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

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

Step 5.4.2.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{2}{- 3}

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

x = \frac{2}{- 3}

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

x = \frac{2}{- 3}

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

Step 5.4.2.3

Simplify the right side.

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Step 5.4.2.3.1

Move the negative in front of the fraction.

x = - \frac{2}{3}

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

x = - \frac{2}{3}

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

x = - \frac{2}{3}

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

x = - \frac{2}{3}

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

x = - \frac{2}{3}

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

Step 6

The domain is all values of

x

$x$ that make the expression defined.

Interval Notation:

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

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

Set-Builder Notation:

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

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

Step 7