Find the Domain and Range (x^2-1)/(x^2+1)

$\frac{x^{2} - 1}{x^{2} + 1}$

Step 1

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

$x^{2} + 1 = 0$

Step 2

Solve for $x$ .

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = i$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i$

The complete solution is the result of both the positive and negative portions of the solution.

$x = i, - i$

Step 3

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4

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

Interval Notation:

$[- 1, 1)$

Set-Builder Notation:

${y | - 1 \leq y < 1}$

Step 5

Determine the domain and range.

Domain: $(- \infty, \infty), {x | x \in ℝ}$

Range: $[- 1, 1), {y | - 1 \leq y < 1}$

Step 6