Calculus Examples

$f (x) = 6 x^{2}$

Step 1

Write $f (x) = 6 x^{2}$ as an equation.

$y = 6 x^{2}$

Step 2

Interchange the variables.

$x = 6 y^{2}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $6 y^{2} = x$ .

$6 y^{2} = x$

Step 3.2

Divide each term in $6 y^{2} = x$ by $6$ and simplify.

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Step 3.2.1

Divide each term in $6 y^{2} = x$ by $6$ .

$\frac{6 y^{2}}{6} = \frac{x}{6}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $6$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{6 y^{2}}{6} = \frac{x}{6}$

Step 3.2.2.1.2

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

$y^{2} = \frac{x}{6}$

Step 3.3

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

$y = \pm \sqrt{\frac{x}{6}}$

Step 3.4

Simplify $\pm \sqrt{\frac{x}{6}}$ .

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Step 3.4.1

Rewrite $\sqrt{\frac{x}{6}}$ as $\frac{\sqrt{x}}{\sqrt{6}}$ .

$y = \pm \frac{\sqrt{x}}{\sqrt{6}}$

Step 3.4.2

Multiply $\frac{\sqrt{x}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$y = \pm \frac{\sqrt{x}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Step 3.4.3

Combine and simplify the denominator.

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Step 3.4.3.1

Multiply $\frac{\sqrt{x}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$y = \pm \frac{\sqrt{x} \sqrt{6}}{\sqrt{6} \sqrt{6}}$

Step 3.4.3.2

Raise $\sqrt{6}$ to the power of $1$ .

$y = \pm \frac{\sqrt{x} \sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}}$

Step 3.4.3.3

Raise $\sqrt{6}$ to the power of $1$ .

$y = \pm \frac{\sqrt{x} \sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}}$

Step 3.4.3.4

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

$y = \pm \frac{\sqrt{x} \sqrt{6}}{{\sqrt{6}}^{1 + 1}}$

Step 3.4.3.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{x} \sqrt{6}}{{\sqrt{6}}^{2}}$

Step 3.4.3.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 3.4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$y = \pm \frac{\sqrt{x} \sqrt{6}}{{(6^{\frac{1}{2}})}^{2}}$

Step 3.4.3.6.2

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

$y = \pm \frac{\sqrt{x} \sqrt{6}}{6^{\frac{1}{2} \cdot 2}}$

Step 3.4.3.6.3

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

$y = \pm \frac{\sqrt{x} \sqrt{6}}{6^{\frac{2}{2}}}$

Step 3.4.3.6.4

Cancel the common factor of $2$ .

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Step 3.4.3.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{x} \sqrt{6}}{6^{\frac{2}{2}}}$

Step 3.4.3.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{x} \sqrt{6}}{6^{1}}$

Step 3.4.3.6.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{x} \sqrt{6}}{6}$

Step 3.4.4

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{x \cdot 6}}{6}$

Step 3.4.5

Reorder factors in $\pm \frac{\sqrt{x \cdot 6}}{6}$ .

$y = \pm \frac{\sqrt{6 x}}{6}$

Step 3.5

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

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Step 3.5.1

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

$y = \frac{\sqrt{6 x}}{6}$

Step 3.5.2

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

$y = - \frac{\sqrt{6 x}}{6}$

Step 3.5.3

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

$y = \frac{\sqrt{6 x}}{6}$

$y = - \frac{\sqrt{6 x}}{6}$

$y = \frac{\sqrt{6 x}}{6}$

$y = - \frac{\sqrt{6 x}}{6}$

$y = \frac{\sqrt{6 x}}{6}$

$y = - \frac{\sqrt{6 x}}{6}$

Step 4

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

$f^{- 1} (x) = \frac{\sqrt{6 x}}{6}, - \frac{\sqrt{6 x}}{6}$

Step 5

Verify if $f^{- 1} (x) = \frac{\sqrt{6 x}}{6}, - \frac{\sqrt{6 x}}{6}$ is the inverse of $f (x) = 6 x^{2}$ .

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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 $f (x) = 6 x^{2}$ and $f^{- 1} (x) = \frac{\sqrt{6 x}}{6}, - \frac{\sqrt{6 x}}{6}$ and compare them.

Step 5.2

Find the range of $f (x) = 6 x^{2}$ .

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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:

$[0, \infty)$

Step 5.3

Find the domain of $\frac{\sqrt{6 x}}{6}$ .

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Step 5.3.1

Set the radicand in $\sqrt{6 x}$ greater than or equal to $0$ to find where the expression is defined.

$6 x \geq 0$

Step 5.3.2

Divide each term in $6 x \geq 0$ by $6$ and simplify.

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Step 5.3.2.1

Divide each term in $6 x \geq 0$ by $6$ .

$\frac{6 x}{6} \geq \frac{0}{6}$

Step 5.3.2.2

Simplify the left side.

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Step 5.3.2.2.1

Cancel the common factor of $6$ .

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Step 5.3.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} \geq \frac{0}{6}$

Step 5.3.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{0}{6}$

Step 5.3.2.3

Simplify the right side.

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Step 5.3.2.3.1

Divide $0$ by $6$ .

$x \geq 0$

Step 5.3.3

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

$[0, \infty)$

Step 5.4

Find the domain of $f (x) = 6 x^{2}$ .

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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 $f^{- 1} (x) = \frac{\sqrt{6 x}}{6}, - \frac{\sqrt{6 x}}{6}$ is the range of $f (x) = 6 x^{2}$ and the range of $f^{- 1} (x) = \frac{\sqrt{6 x}}{6}, - \frac{\sqrt{6 x}}{6}$ is the domain of $f (x) = 6 x^{2}$ , then $f^{- 1} (x) = \frac{\sqrt{6 x}}{6}, - \frac{\sqrt{6 x}}{6}$ is the inverse of $f (x) = 6 x^{2}$ .

$f^{- 1} (x) = \frac{\sqrt{6 x}}{6}, - \frac{\sqrt{6 x}}{6}$

Step 6