Precalculus Examples

$f (x) = \frac{9}{x^{2}}$

Step 1

Write $f (x) = \frac{9}{x^{2}}$ as an equation.

$y = \frac{9}{x^{2}}$

Step 2

Interchange the variables.

$x = \frac{9}{y^{2}}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{9}{y^{2}} = x$ .

$\frac{9}{y^{2}} = x$

Step 3.2

Find the LCD of the terms in the equation.

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Step 3.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y^{2}, 1$

Step 3.2.2

The LCM of one and any expression is the expression.

$y^{2}$

Step 3.3

Multiply each term in $\frac{9}{y^{2}} = x$ by $y^{2}$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $\frac{9}{y^{2}} = x$ by $y^{2}$ .

$\frac{9}{y^{2}} y^{2} = x y^{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $y^{2}$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{9}{y^{2}} y^{2} = x y^{2}$

Step 3.3.2.1.2

Rewrite the expression.

$9 = x y^{2}$

Step 3.4

Solve the equation.

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Step 3.4.1

Rewrite the equation as $x y^{2} = 9$ .

$x y^{2} = 9$

Step 3.4.2

Divide each term in $x y^{2} = 9$ by $x$ and simplify.

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Step 3.4.2.1

Divide each term in $x y^{2} = 9$ by $x$ .

$\frac{x y^{2}}{x} = \frac{9}{x}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $x$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{x y^{2}}{x} = \frac{9}{x}$

Step 3.4.2.2.1.2

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

$y^{2} = \frac{9}{x}$

Step 3.4.3

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

$y = \pm \sqrt{\frac{9}{x}}$

Step 3.4.4

Simplify $\pm \sqrt{\frac{9}{x}}$ .

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Step 3.4.4.1

Rewrite $\sqrt{\frac{9}{x}}$ as $\frac{\sqrt{9}}{\sqrt{x}}$ .

$y = \pm \frac{\sqrt{9}}{\sqrt{x}}$

Step 3.4.4.2

Simplify the numerator.

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Step 3.4.4.2.1

Rewrite $9$ as $3^{2}$ .

$y = \pm \frac{\sqrt{3^{2}}}{\sqrt{x}}$

Step 3.4.4.2.2

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

$y = \pm \frac{3}{\sqrt{x}}$

Step 3.4.4.3

Multiply $\frac{3}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$y = \pm \frac{3}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}$

Step 3.4.4.4

Combine and simplify the denominator.

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Step 3.4.4.4.1

Multiply $\frac{3}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$y = \pm \frac{3 \sqrt{x}}{\sqrt{x} \sqrt{x}}$

Step 3.4.4.4.2

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

$y = \pm \frac{3 \sqrt{x}}{{\sqrt{x}}^{1} \sqrt{x}}$

Step 3.4.4.4.3

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

$y = \pm \frac{3 \sqrt{x}}{{\sqrt{x}}^{1} {\sqrt{x}}^{1}}$

Step 3.4.4.4.4

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

$y = \pm \frac{3 \sqrt{x}}{{\sqrt{x}}^{1 + 1}}$

Step 3.4.4.4.5

Add $1$ and $1$ .

$y = \pm \frac{3 \sqrt{x}}{{\sqrt{x}}^{2}}$

Step 3.4.4.4.6

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

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Step 3.4.4.4.6.1

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

$y = \pm \frac{3 \sqrt{x}}{{(x^{\frac{1}{2}})}^{2}}$

Step 3.4.4.4.6.2

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

$y = \pm \frac{3 \sqrt{x}}{x^{\frac{1}{2} \cdot 2}}$

Step 3.4.4.4.6.3

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

$y = \pm \frac{3 \sqrt{x}}{x^{\frac{2}{2}}}$

Step 3.4.4.4.6.4

Cancel the common factor of $2$ .

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Step 3.4.4.4.6.4.1

Cancel the common factor.

$y = \pm \frac{3 \sqrt{x}}{x^{\frac{2}{2}}}$

Step 3.4.4.4.6.4.2

Rewrite the expression.

$y = \pm \frac{3 \sqrt{x}}{x^{1}}$

Step 3.4.4.4.6.5

Simplify.

$y = \pm \frac{3 \sqrt{x}}{x}$

Step 3.4.5

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

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Step 3.4.5.1

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

$y = \frac{3 \sqrt{x}}{x}$

Step 3.4.5.2

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

$y = - \frac{3 \sqrt{x}}{x}$

Step 3.4.5.3

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

$y = \frac{3 \sqrt{x}}{x}$

$y = - \frac{3 \sqrt{x}}{x}$

$y = \frac{3 \sqrt{x}}{x}$

$y = - \frac{3 \sqrt{x}}{x}$

$y = \frac{3 \sqrt{x}}{x}$

$y = - \frac{3 \sqrt{x}}{x}$

$y = \frac{3 \sqrt{x}}{x}$

$y = - \frac{3 \sqrt{x}}{x}$

Step 4

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

$f^{- 1} (x) = \frac{3 \sqrt{x}}{x}, - \frac{3 \sqrt{x}}{x}$

Step 5

Verify if $f^{- 1} (x) = \frac{3 \sqrt{x}}{x}, - \frac{3 \sqrt{x}}{x}$ is the inverse of $f (x) = \frac{9}{x^{2}}$ .

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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 $f (x) = \frac{9}{x^{2}}$ and $f^{- 1} (x) = \frac{3 \sqrt{x}}{x}, - \frac{3 \sqrt{x}}{x}$ and compare them.

Step 5.2

Find the range of $f (x) = \frac{9}{x^{2}}$ .

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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 $\frac{3 \sqrt{x}}{x}$ .

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Step 5.3.1

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

$x \geq 0$

Step 5.3.2

Set the denominator in $\frac{3 \sqrt{x}}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

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 $f (x) = \frac{9}{x^{2}}$ .

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Step 5.4.1

Set the denominator in $\frac{9}{x^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} = 0$

Step 5.4.2

Solve for $x$ .

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Step 5.4.2.1

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

$x = \pm \sqrt{0}$

Step 5.4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 5.4.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 5.4.2.2.2

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

$x = \pm 0$

Step 5.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 5.4.3

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

$(- \infty, 0) \cup (0, \infty)$

Step 5.5

Since the domain of $f^{- 1} (x) = \frac{3 \sqrt{x}}{x}, - \frac{3 \sqrt{x}}{x}$ is the range of $f (x) = \frac{9}{x^{2}}$ and the range of $f^{- 1} (x) = \frac{3 \sqrt{x}}{x}, - \frac{3 \sqrt{x}}{x}$ is the domain of $f (x) = \frac{9}{x^{2}}$ , then $f^{- 1} (x) = \frac{3 \sqrt{x}}{x}, - \frac{3 \sqrt{x}}{x}$ is the inverse of $f (x) = \frac{9}{x^{2}}$ .

$f^{- 1} (x) = \frac{3 \sqrt{x}}{x}, - \frac{3 \sqrt{x}}{x}$

Step 6