Precalculus Examples

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

Step 1

Write $h (x) = - \frac{4}{x^{2}}$ as an equation.

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

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 $- \frac{4}{y^{2}} = x$ by $y^{2}$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $- \frac{4}{y^{2}} = x$ by $y^{2}$ .

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

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

Move the leading negative in $- \frac{4}{y^{2}}$ into the numerator.

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

Step 3.3.2.1.2

Cancel the common factor.

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

Step 3.3.2.1.3

Rewrite the expression.

$- 4 = x y^{2}$

Step 3.4

Solve the equation.

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Step 3.4.1

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

$x y^{2} = - 4$

Step 3.4.2

Divide each term in $x y^{2} = - 4$ by $x$ and simplify.

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Step 3.4.2.1

Divide each term in $x y^{2} = - 4$ by $x$ .

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

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.

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

Step 3.4.2.2.1.2

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

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

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Move the negative in front of the fraction.

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

Step 3.4.3

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

$y = \pm \sqrt{- \frac{4}{x}}$

Step 3.4.4

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

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Step 3.4.4.1

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

$y = \sqrt{- \frac{4}{x}}$

Step 3.4.4.2

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

$y = - \sqrt{- \frac{4}{x}}$

Step 3.4.4.3

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

$y = \sqrt{- \frac{4}{x}}$

$y = - \sqrt{- \frac{4}{x}}$

$y = \sqrt{- \frac{4}{x}}$

$y = - \sqrt{- \frac{4}{x}}$

$y = \sqrt{- \frac{4}{x}}$

$y = - \sqrt{- \frac{4}{x}}$

$y = \sqrt{- \frac{4}{x}}$

$y = - \sqrt{- \frac{4}{x}}$

Step 4

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

$h^{- 1} (x) = \sqrt{- \frac{4}{x}}, - \sqrt{- \frac{4}{x}}$

Step 5

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

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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 $h (x) = - \frac{4}{x^{2}}$ and $h^{- 1} (x) = \sqrt{- \frac{4}{x}}, - \sqrt{- \frac{4}{x}}$ and compare them.

Step 5.2

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

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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:

$(- \infty, 0)$

Step 5.3

Find the domain of $\sqrt{- \frac{4}{x}}$ .

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Step 5.3.1

Set the radicand in $\sqrt{- \frac{4}{x}}$ greater than or equal to $0$ to find where the expression is defined.

$- \frac{4}{x} \geq 0$

Step 5.3.2

Solve for $x$ .

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Step 5.3.2.1

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$x = 0$

Step 5.3.2.2

Find the domain of $- \frac{4}{x}$ .

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Step 5.3.2.2.1

Set the denominator in $\frac{4}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 5.3.2.2.2

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

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

Step 5.3.2.3

The solution consists of all of the true intervals.

$x < 0$

Step 5.3.3

Set the denominator in $\frac{4}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 5.3.4

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

$(- \infty, 0)$

Step 5.4

Find the domain of $h (x) = - \frac{4}{x^{2}}$ .

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Step 5.4.1

Set the denominator in $\frac{4}{x^{2}}$ 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 $h^{- 1} (x) = \sqrt{- \frac{4}{x}}, - \sqrt{- \frac{4}{x}}$ is the range of $h (x) = - \frac{4}{x^{2}}$ and the range of $h^{- 1} (x) = \sqrt{- \frac{4}{x}}, - \sqrt{- \frac{4}{x}}$ is the domain of $h (x) = - \frac{4}{x^{2}}$ , then $h^{- 1} (x) = \sqrt{- \frac{4}{x}}, - \sqrt{- \frac{4}{x}}$ is the inverse of $h (x) = - \frac{4}{x^{2}}$ .

$h^{- 1} (x) = \sqrt{- \frac{4}{x}}, - \sqrt{- \frac{4}{x}}$

Step 6