Find the Domain and Range y=2+(x^2)/(x^2+4)

$y = 2 + \frac{x^{2}}{x^{2} + 4}$

Set the denominator in $\frac{x^{2}}{x^{2} + 4}$ equal to $0$ to find where the expression is undefined.

$x^{2} + 4 = 0$

Solve for $x$ .

Subtract $4$ from both sides of the equation.

$x^{2} = - 4$

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

$x = \pm \sqrt{- 4}$

Simplify $\pm \sqrt{- 4}$ .

Rewrite $- 4$ as $- 1 (4)$ .

$x = \pm \sqrt{- 1 (4)}$

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{4}$

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

$x = \pm i \cdot \sqrt{4}$

Rewrite $4$ as $2^{2}$ .

$x = \pm i \cdot \sqrt{2^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm i \cdot 2$

Move $2$ to the left of $i$ .

$x = \pm 2 i$

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.

$x = 2 i$

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

$x = - 2 i$

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

$x = 2 i, - 2 i$

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

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

Interval Notation:

$[2, 3)$

Set-Builder Notation:

${y | 2 \leq y < 3}$

Determine the domain and range.

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

Range: $[2, 3), {y | 2 \leq y < 3}$