Find the Domain and Range f(x)=1/( square root of x^2-4)

$f (x) = \frac{1}{\sqrt{x^{2} - 4}}$

Step 1

Set the radicand in $\sqrt{x^{2} - 4}$ greater than or equal to $0$ to find where the expression is defined.

$x^{2} - 4 \geq 0$

Step 2

Solve for $x$ .

Add $4$ to both sides of the inequality.

$x^{2} \geq 4$

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

$\sqrt{x^{2}} \geq \sqrt{4}$

Simplify the equation.

Simplify the left side.

Pull terms out from under the radical.

$| x | \geq \sqrt{4}$

Simplify the right side.

Simplify $\sqrt{4}$ .

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

$| x | \geq \sqrt{2^{2}}$

Pull terms out from under the radical.

$| x | \geq | 2 |$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$| x | \geq 2$

Write $| x | \geq 2$ as a piecewise.

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

In the piece where $x$ is non-negative, remove the absolute value.

$x \geq 2$

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x \geq 2$

Write as a piecewise.

${\begin{cases} x \geq 2 & x \geq 0 \\ - x \geq 2 & x < 0 \end{cases}$

Find the intersection of $x \geq 2$ and $x \geq 0$ .

$x \geq 2$

Divide each term in $- x \geq 2$ by $- 1$ and simplify.

Divide each term in $- x \geq 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{2}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{2}{- 1}$

Divide $x$ by $1$ .

$x \leq \frac{2}{- 1}$

Simplify the right side.

Divide $2$ by $- 1$ .

$x \leq - 2$

Find the union of the solutions.

$x \leq - 2$ or $x \geq 2$

Step 3

Set the denominator in $\frac{1}{\sqrt{x^{2} - 4}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{x^{2} - 4} = 0$

Step 4

Solve for $x$ .

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x^{2} - 4}}^{2} = 0^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} - 4}$ as ${(x^{2} - 4)}^{\frac{1}{2}}$ .

${({(x^{2} - 4)}^{\frac{1}{2}})}^{2} = 0^{2}$

Simplify the left side.

Simplify ${({(x^{2} - 4)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x^{2} - 4)}^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x^{2} - 4)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x^{2} - 4)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Rewrite the expression.

${(x^{2} - 4)}^{1} = 0^{2}$

Simplify.

$x^{2} - 4 = 0^{2}$

Simplify the right side.

Raising $0$ to any positive power yields $0$ .

$x^{2} - 4 = 0$

Solve for $x$ .

Add $4$ to 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 $2^{2}$ .

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

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

$x = \pm 2$

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.

$x = 2$

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

$x = - 2$

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

$x = 2, - 2$

Step 5

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

Interval Notation:

$(- \infty, - 2) \cup (2, \infty)$

Set-Builder Notation:

${x | x < - 2, x > 2}$

Step 6

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

Interval Notation:

$(0, \infty)$

Set-Builder Notation:

${y | y > 0}$

Step 7

Determine the domain and range.

Domain: $(- \infty, - 2) \cup (2, \infty), {x | x < - 2, x > 2}$

Range: $(0, \infty), {y | y > 0}$

Step 8

Find the Domain and Range f(x)=1/( square root of x^2-4)

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