Precalculus Examples

$f (x) = \frac{7}{x} - 3$

Step 1

Write $f (x) = \frac{7}{x} - 3$ as an equation.

$y = \frac{7}{x} - 3$

Step 2

Interchange the variables.

$x = \frac{7}{y} - 3$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{7}{y} - 3 = x$ .

$\frac{7}{y} - 3 = x$

Step 3.2

Add $3$ to both sides of the equation.

$\frac{7}{y} = x + 3$

Step 3.3

Find the LCD of the terms in the equation.

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Step 3.3.1

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

$y, 1, 1$

Step 3.3.2

The LCM of one and any expression is the expression.

$y$

Step 3.4

Multiply each term in $\frac{7}{y} = x + 3$ by $y$ to eliminate the fractions.

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Step 3.4.1

Multiply each term in $\frac{7}{y} = x + 3$ by $y$ .

$\frac{7}{y} y = x y + 3 y$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $y$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{7}{y} y = x y + 3 y$

Step 3.4.2.1.2

Rewrite the expression.

$7 = x y + 3 y$

Step 3.5

Solve the equation.

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Step 3.5.1

Rewrite the equation as $x y + 3 y = 7$ .

$x y + 3 y = 7$

Step 3.5.2

Factor $y$ out of $x y + 3 y$ .

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Step 3.5.2.1

Factor $y$ out of $x y$ .

$y x + 3 y = 7$

Step 3.5.2.2

Factor $y$ out of $3 y$ .

$y x + y \cdot 3 = 7$

Step 3.5.2.3

Factor $y$ out of $y x + y \cdot 3$ .

$y (x + 3) = 7$

Step 3.5.3

Divide each term in $y (x + 3) = 7$ by $x + 3$ and simplify.

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Step 3.5.3.1

Divide each term in $y (x + 3) = 7$ by $x + 3$ .

$\frac{y (x + 3)}{x + 3} = \frac{7}{x + 3}$

Step 3.5.3.2

Simplify the left side.

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Step 3.5.3.2.1

Cancel the common factor of $x + 3$ .

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Step 3.5.3.2.1.1

Cancel the common factor.

$\frac{y (x + 3)}{x + 3} = \frac{7}{x + 3}$

Step 3.5.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{7}{x + 3}$

Step 4

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

$f^{- 1} (x) = \frac{7}{x + 3}$

Step 5

Verify if $f^{- 1} (x) = \frac{7}{x + 3}$ is the inverse of $f (x) = \frac{7}{x} - 3$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (\frac{7}{x} - 3)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{7}{x} - 3) = \frac{7}{(\frac{7}{x} - 3) + 3}$

Step 5.2.3

Simplify the denominator.

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Step 5.2.3.1

Add $- 3$ and $3$ .

$f^{- 1} (\frac{7}{x} - 3) = \frac{7}{\frac{7}{x} + 0}$

Step 5.2.3.2

Add $\frac{7}{x}$ and $0$ .

$f^{- 1} (\frac{7}{x} - 3) = \frac{7}{\frac{7}{x}}$

Step 5.2.4

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{7}{x} - 3) = 7 (\frac{x}{7})$

Step 5.2.5

Cancel the common factor of $7$ .

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Step 5.2.5.1

Cancel the common factor.

$f^{- 1} (\frac{7}{x} - 3) = 7 (\frac{x}{7})$

Step 5.2.5.2

Rewrite the expression.

$f^{- 1} (\frac{7}{x} - 3) = x$

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (\frac{7}{x + 3})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{7}{x + 3}) = \frac{7}{\frac{7}{x + 3}} - 3$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{7}{x + 3}) = 7 (\frac{x + 3}{7}) - 3$

Step 5.3.3.2

Cancel the common factor of $7$ .

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Step 5.3.3.2.1

Cancel the common factor.

$f (\frac{7}{x + 3}) = 7 (\frac{x + 3}{7}) - 3$

Step 5.3.3.2.2

Rewrite the expression.

$f (\frac{7}{x + 3}) = x + 3 - 3$

Step 5.3.4

Combine the opposite terms in $x + 3 - 3$ .

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Step 5.3.4.1

Subtract $3$ from $3$ .

$f (\frac{7}{x + 3}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (\frac{7}{x + 3}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{7}{x + 3}$ is the inverse of $f (x) = \frac{7}{x} - 3$ .

$f^{- 1} (x) = \frac{7}{x + 3}$