Algebra Examples

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

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\sqrt{y^{2} - 4} = x$ .

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

Step 2.2

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

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

Step 2.3

Simplify each side of the equation.

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Step 2.3.1

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

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

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

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Step 2.3.2.1.1

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

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Step 2.3.2.1.1.1

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

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

Step 2.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.3.2.1.1.2.1

Cancel the common factor.

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

Step 2.3.2.1.1.2.2

Rewrite the expression.

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

Step 2.3.2.1.2

Simplify.

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

Step 2.4

Solve for $y$ .

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Step 2.4.1

Add $4$ to both sides of the equation.

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

Step 2.4.2

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

$y = \pm \sqrt{x^{2} + 4}$

Step 2.4.3

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

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Step 2.4.3.1

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

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

Step 2.4.3.2

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

$y = - \sqrt{x^{2} + 4}$

Step 2.4.3.3

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

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

$y = - \sqrt{x^{2} + 4}$

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

$y = - \sqrt{x^{2} + 4}$

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

$y = - \sqrt{x^{2} + 4}$

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

$y = - \sqrt{x^{2} + 4}$

Step 3

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

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

Step 4

Verify if $f^{- 1} (x) = \sqrt{x^{2} + 4}, - \sqrt{x^{2} + 4}$ is the inverse of $f (x) = \sqrt{x^{2} - 4}$ .

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Step 4.1

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = \sqrt{x^{2} - 4}$ and $f^{- 1} (x) = \sqrt{x^{2} + 4}, - \sqrt{x^{2} + 4}$ and compare them.

Step 4.2

Find the range of $f (x) = \sqrt{x^{2} - 4}$ .

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Step 4.2.1

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

Interval Notation:

$[0, \infty)$

Step 4.3

Find the domain of $\sqrt{x^{2} + 4}$ .

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Step 4.3.1

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 4.3.2

Solve for $x$ .

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Step 4.3.2.1

Subtract $4$ from both sides of the inequality.

$x^{2} \geq - 4$

Step 4.3.2.2

Since the left side has an even power, it is always positive for all real numbers.

All real numbers

Step 4.3.3

The domain is all real numbers.

$(- \infty, \infty)$

Step 4.4

Since the domain of $f^{- 1} (x) = \sqrt{x^{2} + 4}, - \sqrt{x^{2} + 4}$ is not equal to the range of $f (x) = \sqrt{x^{2} - 4}$ , then $f^{- 1} (x) = \sqrt{x^{2} + 4}, - \sqrt{x^{2} + 4}$ is not an inverse of $f (x) = \sqrt{x^{2} - 4}$ .

There is no inverse

Step 5