Find the Domain and Range 1/(x^2+64)+4

$\frac{1}{x^{2} + 64} + 4$

Step 1

Set the denominator in $\frac{1}{x^{2} + 64}$ equal to $0$ to find where the expression is undefined.

$x^{2} + 64 = 0$

Step 2

Solve for $x$ .

Subtract $64$ from both sides of the equation.

$x^{2} = - 64$

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

$x = \pm \sqrt{- 64}$

Simplify $\pm \sqrt{- 64}$ .

Rewrite $- 64$ as $- 1 (64)$ .

$x = \pm \sqrt{- 1 (64)}$

Rewrite $\sqrt{- 1 (64)}$ as $\sqrt{- 1} \cdot \sqrt{64}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{64}$

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

$x = \pm i \cdot \sqrt{64}$

Rewrite $64$ as $8^{2}$ .

$x = \pm i \cdot \sqrt{8^{2}}$

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

$x = \pm i \cdot 8$

Move $8$ to the left of $i$ .

$x = \pm 8 i$

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.

$x = 8 i$

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

$x = - 8 i$

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

$x = 8 i, - 8 i$

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:

$(4, \frac{257}{64}]$

Set-Builder Notation:

${y | 4 < y \leq \frac{257}{64}}$

Step 5

Determine the domain and range.

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

Range: $(4, \frac{257}{64}], {y | 4 < y \leq \frac{257}{64}}$

Step 6

Find the Domain and Range 1/(x^2+64)+4

Related Questions

Support LCMGCF.com by disabling your adblocker.