Algebra Examples

$f (x) = 5 x^{2} + 2$ , $x \geq 0$

Step 1

Find the range of the given function.

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Step 1.1

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

$[2, \infty)$

Step 1.2

Convert $[2, \infty)$ to an inequality.

$y \geq 2$

Step 2

Find the inverse.

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Step 2.1

Interchange the variables.

$x = 5 y^{2} + 2$

Step 2.2

Solve for $y$ .

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Step 2.2.1

Rewrite the equation as $5 y^{2} + 2 = x$ .

$5 y^{2} + 2 = x$

Step 2.2.2

Subtract $2$ from both sides of the equation.

$5 y^{2} = x - 2$

Step 2.2.3

Divide each term in $5 y^{2} = x - 2$ by $5$ and simplify.

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Step 2.2.3.1

Divide each term in $5 y^{2} = x - 2$ by $5$ .

$\frac{5 y^{2}}{5} = \frac{x}{5} + \frac{- 2}{5}$

Step 2.2.3.2

Simplify the left side.

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Step 2.2.3.2.1

Cancel the common factor of $5$ .

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Step 2.2.3.2.1.1

Cancel the common factor.

$\frac{5 y^{2}}{5} = \frac{x}{5} + \frac{- 2}{5}$

Step 2.2.3.2.1.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{x}{5} + \frac{- 2}{5}$

Step 2.2.3.3

Simplify the right side.

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Step 2.2.3.3.1

Move the negative in front of the fraction.

$y^{2} = \frac{x}{5} - \frac{2}{5}$

Step 2.2.4

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

$y = \pm \sqrt{\frac{x}{5} - \frac{2}{5}}$

Step 2.2.5

Simplify $\pm \sqrt{\frac{x}{5} - \frac{2}{5}}$ .

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Step 2.2.5.1

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{x - 2}{5}}$

Step 2.2.5.2

Rewrite $\sqrt{\frac{x - 2}{5}}$ as $\frac{\sqrt{x - 2}}{\sqrt{5}}$ .

$y = \pm \frac{\sqrt{x - 2}}{\sqrt{5}}$

Step 2.2.5.3

Multiply $\frac{\sqrt{x - 2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$y = \pm \frac{\sqrt{x - 2}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 2.2.5.4

Combine and simplify the denominator.

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Step 2.2.5.4.1

Multiply $\frac{\sqrt{x - 2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$y = \pm \frac{\sqrt{x - 2} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 2.2.5.4.2

Raise $\sqrt{5}$ to the power of $1$ .

$y = \pm \frac{\sqrt{x - 2} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 2.2.5.4.3

Raise $\sqrt{5}$ to the power of $1$ .

$y = \pm \frac{\sqrt{x - 2} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 2.2.5.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \pm \frac{\sqrt{x - 2} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 2.2.5.4.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{x - 2} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 2.2.5.4.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 2.2.5.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$y = \pm \frac{\sqrt{x - 2} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 2.2.5.4.6.2

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

$y = \pm \frac{\sqrt{x - 2} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 2.2.5.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$y = \pm \frac{\sqrt{x - 2} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 2.2.5.4.6.4

Cancel the common factor of $2$ .

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Step 2.2.5.4.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{x - 2} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 2.2.5.4.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{x - 2} \sqrt{5}}{5^{1}}$

Step 2.2.5.4.6.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{x - 2} \sqrt{5}}{5}$

Step 2.2.5.5

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{(x - 2) \cdot 5}}{5}$

Step 2.2.5.6

Reorder factors in $\pm \frac{\sqrt{(x - 2) \cdot 5}}{5}$ .

$y = \pm \frac{\sqrt{5 (x - 2)}}{5}$

Step 2.2.6

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

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Step 2.2.6.1

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

$y = \frac{\sqrt{5 (x - 2)}}{5}$

Step 2.2.6.2

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

$y = - \frac{\sqrt{5 (x - 2)}}{5}$

Step 2.2.6.3

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

$y = \frac{\sqrt{5 (x - 2)}}{5}$

$y = - \frac{\sqrt{5 (x - 2)}}{5}$

$y = \frac{\sqrt{5 (x - 2)}}{5}$

$y = - \frac{\sqrt{5 (x - 2)}}{5}$

$y = \frac{\sqrt{5 (x - 2)}}{5}$

$y = - \frac{\sqrt{5 (x - 2)}}{5}$

Step 2.3

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

$f^{- 1} (x) = \frac{\sqrt{5 (x - 2)}}{5}, - \frac{\sqrt{5 (x - 2)}}{5}$

Step 3

Find the inverse using the domain and the range of the original function.

$f^{- 1} (x) = \frac{\sqrt{5 (x - 2)}}{5}, x \geq 2$

Step 4