Algebra Examples

f (x) = 2 x^{2} - 8

$f (x) = 2 x^{2} - 8$

Step 1

Write

f (x) = 2 x^{2} - 8

$f (x) = 2 x^{2} - 8$ as an equation.

y = 2 x^{2} - 8

$y = 2 x^{2} - 8$

Step 2

Interchange the variables.

x = 2 y^{2} - 8

$x = 2 y^{2} - 8$

Step 3

Solve for

y

$y$ .

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Step 3.1

Rewrite the equation as

2 y^{2} - 8 = x

$2 y^{2} - 8 = x$ .

2 y^{2} - 8 = x

$2 y^{2} - 8 = x$

Step 3.2

Add

8

$8$ to both sides of the equation.

2 y^{2} = x + 8

$2 y^{2} = x + 8$

Step 3.3

Divide each term in

2 y^{2} = x + 8

$2 y^{2} = x + 8$ by

2

$2$ and simplify.

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Step 3.3.1

Divide each term in

2 y^{2} = x + 8

$2 y^{2} = x + 8$ by

2

$2$ .

\frac{2 y^{2}}{2} = \frac{x}{2} + \frac{8}{2}

$\frac{2 y^{2}}{2} = \frac{x}{2} + \frac{8}{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of

2

$2$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{2 y^{2}}{2} = \frac{x}{2} + \frac{8}{2}$

Step 3.3.2.1.2

Divide

$y^{2}$ by

$1$ .

$y^{2} = \frac{x}{2} + \frac{8}{2}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide

$8$ by

$2$ .

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

Step 3.4

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

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

Step 3.5

Simplify

$\pm \sqrt{\frac{x}{2} + 4}$ .

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Step 3.5.1

To write

$4$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

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

Step 3.5.2

Combine

$4$ and

$\frac{2}{2}$ .

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

Step 3.5.3

Combine the numerators over the common denominator.

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

Step 3.5.4

Multiply

$4$ by

$2$ .

$y = \pm \sqrt{\frac{x + 8}{2}}$

Step 3.5.5

Rewrite

$\sqrt{\frac{x + 8}{2}}$ as

$\frac{\sqrt{x + 8}}{\sqrt{2}}$ .

$y = \pm \frac{\sqrt{x + 8}}{\sqrt{2}}$

Step 3.5.6

Multiply

$\frac{\sqrt{x + 8}}{\sqrt{2}}$ by

$\frac{\sqrt{2}}{\sqrt{2}}$ .

$y = \pm \frac{\sqrt{x + 8}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 3.5.7

Combine and simplify the denominator.

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Step 3.5.7.1

Multiply

$\frac{\sqrt{x + 8}}{\sqrt{2}}$ by

$\frac{\sqrt{2}}{\sqrt{2}}$ .

$y = \pm \frac{\sqrt{x + 8} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 3.5.7.2

Raise

$\sqrt{2}$ to the power of

$1$ .

$y = \pm \frac{\sqrt{x + 8} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 3.5.7.3

Raise

$\sqrt{2}$ to the power of

$1$ .

$y = \pm \frac{\sqrt{x + 8} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 3.5.7.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \pm \frac{\sqrt{x + 8} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 3.5.7.5

Add

$1$ and

$1$ .

$y = \pm \frac{\sqrt{x + 8} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 3.5.7.6

Rewrite

${\sqrt{2}}^{2}$ as

$2$ .

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Step 3.5.7.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2}$ as

$2^{\frac{1}{2}}$ .

$y = \pm \frac{\sqrt{x + 8} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 3.5.7.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$y = \pm \frac{\sqrt{x + 8} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 3.5.7.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$y = \pm \frac{\sqrt{x + 8} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 3.5.7.6.4

Cancel the common factor of

$2$ .

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Step 3.5.7.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{x + 8} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 3.5.7.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{x + 8} \sqrt{2}}{2^{1}}$

Step 3.5.7.6.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{x + 8} \sqrt{2}}{2}$

Step 3.5.8

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{(x + 8) \cdot 2}}{2}$

Step 3.5.9

Reorder factors in

$\pm \frac{\sqrt{(x + 8) \cdot 2}}{2}$ .

$y = \pm \frac{\sqrt{2 (x + 8)}}{2}$

Step 3.6

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

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Step 3.6.1

First, use the positive value of the

$\pm$ to find the first solution.

$y = \frac{\sqrt{2 (x + 8)}}{2}$

Step 3.6.2

Next, use the negative value of the

$\pm$ to find the second solution.

$y = - \frac{\sqrt{2 (x + 8)}}{2}$

Step 3.6.3

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

$y = \frac{\sqrt{2 (x + 8)}}{2}$

$y = - \frac{\sqrt{2 (x + 8)}}{2}$

$y = \frac{\sqrt{2 (x + 8)}}{2}$

$y = - \frac{\sqrt{2 (x + 8)}}{2}$

$y = \frac{\sqrt{2 (x + 8)}}{2}$

$y = - \frac{\sqrt{2 (x + 8)}}{2}$

Step 4

Replace

$y$ with

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

$f^{- 1} (x) = \frac{\sqrt{2 (x + 8)}}{2}, - \frac{\sqrt{2 (x + 8)}}{2}$

Step 5

Verify if

$f^{- 1} (x) = \frac{\sqrt{2 (x + 8)}}{2}, - \frac{\sqrt{2 (x + 8)}}{2}$ is the inverse of

$f (x) = 2 x^{2} - 8$ .

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Step 5.1

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of

$f (x) = 2 x^{2} - 8$ and

$f^{- 1} (x) = \frac{\sqrt{2 (x + 8)}}{2}, - \frac{\sqrt{2 (x + 8)}}{2}$ and compare them.

Step 5.2

Find the range of

$f (x) = 2 x^{2} - 8$ .

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Step 5.2.1

The range is the set of all valid

$y$ values. Use the graph to find the range.

Interval Notation:

$[- 8, \infty)$

Step 5.3

Find the domain of

$\frac{\sqrt{2 (x + 8)}}{2}$ .

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Step 5.3.1

Set the radicand in

$\sqrt{2 (x + 8)}$ greater than or equal to

$0$ to find where the expression is defined.

$2 (x + 8) \geq 0$

Step 5.3.2

Solve for

$x$ .

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Step 5.3.2.1

Divide each term in

$2 (x + 8) \geq 0$ by

$2$ and simplify.

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Step 5.3.2.1.1

Divide each term in

$2 (x + 8) \geq 0$ by

$2$ .

$\frac{2 (x + 8)}{2} \geq \frac{0}{2}$

Step 5.3.2.1.2

Simplify the left side.

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Step 5.3.2.1.2.1

Cancel the common factor of

$2$ .

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Step 5.3.2.1.2.1.1

Cancel the common factor.

$\frac{2 (x + 8)}{2} \geq \frac{0}{2}$

Step 5.3.2.1.2.1.2

Divide

$x + 8$ by

$1$ .

$x + 8 \geq \frac{0}{2}$

Step 5.3.2.1.3

Simplify the right side.

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Step 5.3.2.1.3.1

Divide

$0$ by

$2$ .

$x + 8 \geq 0$

Step 5.3.2.2

Subtract

$8$ from both sides of the inequality.

$x \geq - 8$

Step 5.3.3

The domain is all values of

$x$ that make the expression defined.

$[- 8, \infty)$

Step 5.4

Find the domain of

$f (x) = 2 x^{2} - 8$ .

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Step 5.4.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

$(- \infty, \infty)$

Step 5.5

Since the domain of

$f^{- 1} (x) = \frac{\sqrt{2 (x + 8)}}{2}, - \frac{\sqrt{2 (x + 8)}}{2}$ is the range of

$f (x) = 2 x^{2} - 8$ and the range of

$f^{- 1} (x) = \frac{\sqrt{2 (x + 8)}}{2}, - \frac{\sqrt{2 (x + 8)}}{2}$ is the domain of

$f (x) = 2 x^{2} - 8$ , then

$f^{- 1} (x) = \frac{\sqrt{2 (x + 8)}}{2}, - \frac{\sqrt{2 (x + 8)}}{2}$ is the inverse of

$f (x) = 2 x^{2} - 8$ .

$f^{- 1} (x) = \frac{\sqrt{2 (x + 8)}}{2}, - \frac{\sqrt{2 (x + 8)}}{2}$

Step 6