Precalculus Examples

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

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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Step 2.1

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

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

Step 2.2

Find the LCD of the terms in the equation.

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Step 2.2.1

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

$y, 1$

Step 2.2.2

The LCM of one and any expression is the expression.

$y$

Step 2.3

Multiply each term in $- \frac{4}{y} = x$ by $y$ to eliminate the fractions.

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Step 2.3.1

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

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $y$ .

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Step 2.3.2.1.1

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

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

Step 2.3.2.1.2

Cancel the common factor.

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

Step 2.3.2.1.3

Rewrite the expression.

$- 4 = x y$

Step 2.4

Solve the equation.

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Step 2.4.1

Rewrite the equation as $x y = - 4$ .

$x y = - 4$

Step 2.4.2

Divide each term in $x y = - 4$ by $x$ and simplify.

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Step 2.4.2.1

Divide each term in $x y = - 4$ by $x$ .

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

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Cancel the common factor of $x$ .

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Step 2.4.2.2.1.1

Cancel the common factor.

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

Step 2.4.2.2.1.2

Divide $y$ by $1$ .

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

Step 2.4.2.3

Simplify the right side.

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Step 2.4.2.3.1

Move the negative in front of the fraction.

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

Step 3

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

$f^{- 1} (x) = - \frac{4}{x}$

Step 4

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

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Step 4.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 4.2

Evaluate $f^{- 1} (f (x))$ .

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Step 4.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 4.2.2

Evaluate $f^{- 1} (- \frac{4}{x})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- \frac{4}{x}) = - \frac{4}{- \frac{4}{x}}$

Step 4.2.3

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (- \frac{4}{x}) = - (4 (- \frac{x}{4}))$

Step 4.2.4

Cancel the common factor of $4$ .

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Step 4.2.4.1

Move the leading negative in $- \frac{x}{4}$ into the numerator.

$f^{- 1} (- \frac{4}{x}) = - (4 (\frac{- x}{4}))$

Step 4.2.4.2

Cancel the common factor.

$f^{- 1} (- \frac{4}{x}) = - (4 (\frac{- x}{4}))$

Step 4.2.4.3

Rewrite the expression.

$f^{- 1} (- \frac{4}{x}) = x$

Step 4.3

Evaluate $f (f^{- 1} (x))$ .

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Step 4.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 4.3.2

Evaluate $f (- \frac{4}{x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{4}{x}) = - \frac{4}{- \frac{4}{x}}$

Step 4.3.3

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{4}{x}) = - (4 (- \frac{x}{4}))$

Step 4.3.4

Cancel the common factor of $4$ .

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Step 4.3.4.1

Move the leading negative in $- \frac{x}{4}$ into the numerator.

$f (- \frac{4}{x}) = - (4 (\frac{- x}{4}))$

Step 4.3.4.2

Cancel the common factor.

$f (- \frac{4}{x}) = - (4 (\frac{- x}{4}))$

Step 4.3.4.3

Rewrite the expression.

$f (- \frac{4}{x}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{4}{x}$ is the inverse of $f (x) = - \frac{4}{x}$ .

$f^{- 1} (x) = - \frac{4}{x}$