Precalculus Examples

$f (x) = \frac{7}{4 x + 12}$

Step 1

Write $f (x) = \frac{7}{4 x + 12}$ as an equation.

$y = \frac{7}{4 x + 12}$

Step 2

Interchange the variables.

$x = \frac{7}{4 y + 12}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{7}{4 y + 12} = x$ .

$\frac{7}{4 y + 12} = x$

Step 3.2

Factor $4$ out of $4 y + 12$ .

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Step 3.2.1

Factor $4$ out of $4 y$ .

$\frac{7}{4 (y) + 12} = x$

Step 3.2.2

Factor $4$ out of $12$ .

$\frac{7}{4 y + 4 \cdot 3} = x$

Step 3.2.3

Factor $4$ out of $4 y + 4 \cdot 3$ .

$\frac{7}{4 (y + 3)} = x$

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.

$4 (y + 3), 1$

Step 3.3.2

The LCM of one and any expression is the expression.

$4 (y + 3)$

Step 3.4

Multiply each term in $\frac{7}{4 (y + 3)} = x$ by $4 (y + 3)$ to eliminate the fractions.

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Step 3.4.1

Multiply each term in $\frac{7}{4 (y + 3)} = x$ by $4 (y + 3)$ .

$\frac{7}{4 (y + 3)} (4 (y + 3)) = x (4 (y + 3))$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Rewrite using the commutative property of multiplication.

$4 \frac{7}{4 (y + 3)} (y + 3) = x (4 (y + 3))$

Step 3.4.2.2

Cancel the common factor of $4$ .

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Step 3.4.2.2.1

Cancel the common factor.

$4 \frac{7}{4 (y + 3)} (y + 3) = x (4 (y + 3))$

Step 3.4.2.2.2

Rewrite the expression.

$\frac{7}{y + 3} (y + 3) = x (4 (y + 3))$

Step 3.4.2.3

Cancel the common factor of $y + 3$ .

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Step 3.4.2.3.1

Cancel the common factor.

$\frac{7}{y + 3} (y + 3) = x (4 (y + 3))$

Step 3.4.2.3.2

Rewrite the expression.

$7 = x (4 (y + 3))$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Rewrite using the commutative property of multiplication.

$7 = 4 x (y + 3)$

Step 3.4.3.2

Apply the distributive property.

$7 = 4 x y + 4 x \cdot 3$

Step 3.4.3.3

Multiply $3$ by $4$ .

$7 = 4 x y + 12 x$

Step 3.5

Solve the equation.

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Step 3.5.1

Rewrite the equation as $4 x y + 12 x = 7$ .

$4 x y + 12 x = 7$

Step 3.5.2

Subtract $12 x$ from both sides of the equation.

$4 x y = 7 - 12 x$

Step 3.5.3

Divide each term in $4 x y = 7 - 12 x$ by $4 x$ and simplify.

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Step 3.5.3.1

Divide each term in $4 x y = 7 - 12 x$ by $4 x$ .

$\frac{4 x y}{4 x} = \frac{7}{4 x} + \frac{- 12 x}{4 x}$

Step 3.5.3.2

Simplify the left side.

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Step 3.5.3.2.1

Cancel the common factor of $4$ .

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Step 3.5.3.2.1.1

Cancel the common factor.

$\frac{4 x y}{4 x} = \frac{7}{4 x} + \frac{- 12 x}{4 x}$

Step 3.5.3.2.1.2

Rewrite the expression.

$\frac{x y}{x} = \frac{7}{4 x} + \frac{- 12 x}{4 x}$

Step 3.5.3.2.2

Cancel the common factor of $x$ .

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Step 3.5.3.2.2.1

Cancel the common factor.

$\frac{x y}{x} = \frac{7}{4 x} + \frac{- 12 x}{4 x}$

Step 3.5.3.2.2.2

Divide $y$ by $1$ .

$y = \frac{7}{4 x} + \frac{- 12 x}{4 x}$

Step 3.5.3.3

Simplify the right side.

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Step 3.5.3.3.1

Simplify each term.

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Step 3.5.3.3.1.1

Cancel the common factor of $- 12$ and $4$ .

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Step 3.5.3.3.1.1.1

Factor $4$ out of $- 12 x$ .

$y = \frac{7}{4 x} + \frac{4 (- 3 x)}{4 x}$

Step 3.5.3.3.1.1.2

Cancel the common factors.

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Step 3.5.3.3.1.1.2.1

Factor $4$ out of $4 x$ .

$y = \frac{7}{4 x} + \frac{4 (- 3 x)}{4 (x)}$

Step 3.5.3.3.1.1.2.2

Cancel the common factor.

$y = \frac{7}{4 x} + \frac{4 (- 3 x)}{4 x}$

Step 3.5.3.3.1.1.2.3

Rewrite the expression.

$y = \frac{7}{4 x} + \frac{- 3 x}{x}$

Step 3.5.3.3.1.2

Cancel the common factor of $x$ .

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Step 3.5.3.3.1.2.1

Cancel the common factor.

$y = \frac{7}{4 x} + \frac{- 3 x}{x}$

Step 3.5.3.3.1.2.2

Divide $- 3$ by $1$ .

$y = \frac{7}{4 x} - 3$

Step 4

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

$f^{- 1} (x) = \frac{7}{4 x} - 3$

Step 5

Verify if $f^{- 1} (x) = \frac{7}{4 x} - 3$ is the inverse of $f (x) = \frac{7}{4 x + 12}$ .

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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 $f^{- 1} (\frac{7}{4 x + 12})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{7}{4 x + 12}) = \frac{7}{4 (\frac{7}{4 x + 12})} - 3$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Factor $4$ out of $4 x + 12$ .

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Step 5.2.3.1.1

Factor $4$ out of $4 x$ .

$f^{- 1} (\frac{7}{4 x + 12}) = \frac{7}{4 (\frac{7}{4 (x) + 12})} - 3$

Step 5.2.3.1.2

Factor $4$ out of $12$ .

$f^{- 1} (\frac{7}{4 x + 12}) = \frac{7}{4 (\frac{7}{4 x + 4 \cdot 3})} - 3$

Step 5.2.3.1.3

Factor $4$ out of $4 x + 4 \cdot 3$ .

$f^{- 1} (\frac{7}{4 x + 12}) = \frac{7}{4 (\frac{7}{4 (x + 3)})} - 3$

Step 5.2.3.2

Combine $4$ and $\frac{7}{4 (x + 3)}$ .

$f^{- 1} (\frac{7}{4 x + 12}) = \frac{7}{\frac{4 \cdot 7}{4 (x + 3)}} - 3$

Step 5.2.3.3

Multiply $4$ by $7$ .

$f^{- 1} (\frac{7}{4 x + 12}) = \frac{7}{\frac{28}{4 (x + 3)}} - 3$

Step 5.2.3.4

Reduce the expression $\frac{28}{4 (x + 3)}$ by cancelling the common factors.

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Step 5.2.3.4.1

Factor $4$ out of $28$ .

$f^{- 1} (\frac{7}{4 x + 12}) = \frac{7}{\frac{4 \cdot 7}{4 (x + 3)}} - 3$

Step 5.2.3.4.2

Cancel the common factor.

$f^{- 1} (\frac{7}{4 x + 12}) = \frac{7}{\frac{4 \cdot 7}{4 (x + 3)}} - 3$

Step 5.2.3.4.3

Rewrite the expression.

$f^{- 1} (\frac{7}{4 x + 12}) = \frac{7}{\frac{7}{x + 3}} - 3$

Step 5.2.3.5

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{7}{4 x + 12}) = 7 (\frac{x + 3}{7}) - 3$

Step 5.2.3.6

Cancel the common factor of $7$ .

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Step 5.2.3.6.1

Cancel the common factor.

$f^{- 1} (\frac{7}{4 x + 12}) = 7 (\frac{x + 3}{7}) - 3$

Step 5.2.3.6.2

Rewrite the expression.

$f^{- 1} (\frac{7}{4 x + 12}) = x + 3 - 3$

Step 5.2.4

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

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Step 5.2.4.1

Subtract $3$ from $3$ .

$f^{- 1} (\frac{7}{4 x + 12}) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

$f^{- 1} (\frac{7}{4 x + 12}) = x$

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 $f (\frac{7}{4 x} - 3)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{7}{4 x} - 3) = \frac{7}{4 (\frac{7}{4 x} - 3) + 12}$

Step 5.3.3

Simplify the denominator.

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Step 5.3.3.1

Factor $4$ out of $4 (\frac{7}{4 x} - 3) + 12$ .

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Step 5.3.3.1.1

Factor $4$ out of $12$ .

$f (\frac{7}{4 x} - 3) = \frac{7}{4 (\frac{7}{4 x} - 3) + 4 \cdot 3}$

Step 5.3.3.1.2

Factor $4$ out of $4 (\frac{7}{4 x} - 3) + 4 \cdot 3$ .

$f (\frac{7}{4 x} - 3) = \frac{7}{4 (\frac{7}{4 x} - 3 + 3)}$

Step 5.3.3.2

Add $- 3$ and $3$ .

$f (\frac{7}{4 x} - 3) = \frac{7}{4 (\frac{7}{4 x} + 0)}$

Step 5.3.3.3

Add $\frac{7}{4 x}$ and $0$ .

$f (\frac{7}{4 x} - 3) = \frac{7}{4 (\frac{7}{4 x})}$

Step 5.3.4

Simplify terms.

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Step 5.3.4.1

Combine $4$ and $\frac{7}{4 x}$ .

$f (\frac{7}{4 x} - 3) = \frac{7}{\frac{4 \cdot 7}{4 x}}$

Step 5.3.4.2

Multiply $4$ by $7$ .

$f (\frac{7}{4 x} - 3) = \frac{7}{\frac{28}{4 x}}$

Step 5.3.4.3

Reduce the expression $\frac{28}{4 x}$ by cancelling the common factors.

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Step 5.3.4.3.1

Factor $4$ out of $28$ .

$f (\frac{7}{4 x} - 3) = \frac{7}{\frac{4 \cdot 7}{4 x}}$

Step 5.3.4.3.2

Factor $4$ out of $4 x$ .

$f (\frac{7}{4 x} - 3) = \frac{7}{\frac{4 \cdot 7}{4 (x)}}$

Step 5.3.4.3.3

Cancel the common factor.

$f (\frac{7}{4 x} - 3) = \frac{7}{\frac{4 \cdot 7}{4 x}}$

Step 5.3.4.3.4

Rewrite the expression.

$f (\frac{7}{4 x} - 3) = \frac{7}{\frac{7}{x}}$

Step 5.3.5

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{7}{4 x} - 3) = 7 (\frac{x}{7})$

Step 5.3.6

Cancel the common factor of $7$ .

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Step 5.3.6.1

Cancel the common factor.

$f (\frac{7}{4 x} - 3) = 7 (\frac{x}{7})$

Step 5.3.6.2

Rewrite the expression.

$f (\frac{7}{4 x} - 3) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{7}{4 x} - 3$ is the inverse of $f (x) = \frac{7}{4 x + 12}$ .

$f^{- 1} (x) = \frac{7}{4 x} - 3$