Algebra Examples

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

Step 1

Cancel the common factor of $x + 1$ and $- x - 1$ .

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Step 1.1

Factor $- 1$ out of $x$ .

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

Step 1.2

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 1.3

Factor $- 1$ out of $- 1 (- x) - 1 (- 1)$ .

$f (x) = \frac{- 1 (- x - 1)}{- x - 1}$

Step 1.4

Cancel the common factor.

$f (x) = \frac{- 1 (- x - 1)}{- x - 1}$

Step 1.5

Divide $- 1$ by $1$ .

$f (x) = - 1$

Step 2

To find the holes in the graph, look at the denominator factors that were cancelled.

$- x - 1$

Step 3

To find the coordinates of the holes, set each factor that was cancelled equal to $0$ , solve, and substitute back in to $- 1$ .

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Step 3.1

Set $- x - 1$ equal to $0$ .

$- x - 1 = 0$

Step 3.2

Solve $- x - 1 = 0$ for $x$ .

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Step 3.2.1

Add $1$ to both sides of the equation.

$- x = 1$

Step 3.2.2

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

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Step 3.2.2.1

Divide each term in $- x = 1$ by $- 1$ .

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

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 3.2.2.2.2

Divide $x$ by $1$ .

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

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Divide $1$ by $- 1$ .

$x = - 1$

Step 3.3

Because the cancelled expression is constant with respect to $x$ , it is the $y$ coordinate for every hole.

$- 1$

Step 3.4

The holes in the graph are the points where any of the cancelled factors are equal to $0$ .

$(- 1, - 1)$

Step 4