Precalculus Examples

$f (u) = \frac{u + 1}{1 + \frac{1}{u + 1}}$

Step 1

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

$u + 1 = 0$

Step 2

Subtract $1$ from both sides of the equation.

$u = - 1$

Step 3

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

$1 + \frac{1}{u + 1} = 0$

Step 4

Solve for $u$ .

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Step 4.1

Subtract $1$ from both sides of the equation.

$\frac{1}{u + 1} = - 1$

Step 4.2

Find the LCD of the terms in the equation.

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Step 4.2.1

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

$u + 1, 1$

Step 4.2.2

Remove parentheses.

$u + 1, 1$

Step 4.2.3

The LCM of one and any expression is the expression.

$u + 1$

Step 4.3

Multiply each term in $\frac{1}{u + 1} = - 1$ by $u + 1$ to eliminate the fractions.

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Step 4.3.1

Multiply each term in $\frac{1}{u + 1} = - 1$ by $u + 1$ .

$\frac{1}{u + 1} (u + 1) = - (u + 1)$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $u + 1$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{1}{u + 1} (u + 1) = - (u + 1)$

Step 4.3.2.1.2

Rewrite the expression.

$1 = - (u + 1)$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Apply the distributive property.

$1 = - u - 1 \cdot 1$

Step 4.3.3.2

Multiply $- 1$ by $1$ .

$1 = - u - 1$

Step 4.4

Solve the equation.

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Step 4.4.1

Rewrite the equation as $- u - 1 = 1$ .

$- u - 1 = 1$

Step 4.4.2

Move all terms not containing $u$ to the right side of the equation.

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Step 4.4.2.1

Add $1$ to both sides of the equation.

$- u = 1 + 1$

Step 4.4.2.2

Add $1$ and $1$ .

$- u = 2$

Step 4.4.3

Divide each term in $- u = 2$ by $- 1$ and simplify.

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Step 4.4.3.1

Divide each term in $- u = 2$ by $- 1$ .

$\frac{- u}{- 1} = \frac{2}{- 1}$

Step 4.4.3.2

Simplify the left side.

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Step 4.4.3.2.1

Dividing two negative values results in a positive value.

$\frac{u}{1} = \frac{2}{- 1}$

Step 4.4.3.2.2

Divide $u$ by $1$ .

$u = \frac{2}{- 1}$

Step 4.4.3.3

Simplify the right side.

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Step 4.4.3.3.1

Divide $2$ by $- 1$ .

$u = - 2$

Step 5

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

Interval Notation:

$(- \infty, - 2) \cup (- 2, - 1) \cup (- 1, \infty)$

Set-Builder Notation:

${u | u \neq - 1, - 2}$

Step 6