Calculus Examples

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

Step 1

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

$1 - x = 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $1$ from both sides of the equation.

$- x = - 1$

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Dividing two negative values results in a positive value.

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

Step 2.2.2.2

Divide $x$ by $1$ .

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

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $- 1$ by $- 1$ .

$x = 1$

Step 3

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

Interval Notation:

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

Set-Builder Notation:

${x | x \neq 1}$

Step 4

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

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

Set-Builder Notation:

${y | y \neq - 1}$

Step 5

Determine the domain and range.

Domain: $(- \infty, 1) \cup (1, \infty), {x | x \neq 1}$

Range: $(- \infty, - 1) \cup (- 1, \infty), {y | y \neq - 1}$

Step 6