Calculus Examples

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

Step 1

Write $f (x) = \frac{3}{x - 4}$ as an equation.

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{3}{y - 4} = x$ .

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

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.

$y - 4, 1$

Step 3.2.2

Remove parentheses.

$y - 4, 1$

Step 3.2.3

The LCM of one and any expression is the expression.

$y - 4$

Step 3.3

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

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Step 3.3.1

Multiply each term in $\frac{3}{y - 4} = x$ by $y - 4$ .

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $y - 4$ .

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Step 3.3.2.1.1

Cancel the common factor.

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

Step 3.3.2.1.2

Rewrite the expression.

$3 = x (y - 4)$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Apply the distributive property.

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

Step 3.3.3.2

Move $- 4$ to the left of $x$ .

$3 = x y - 4 x$

Step 3.4

Solve the equation.

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Step 3.4.1

Rewrite the equation as $x y - 4 x = 3$ .

$x y - 4 x = 3$

Step 3.4.2

Add $4 x$ to both sides of the equation.

$x y = 3 + 4 x$

Step 3.4.3

Divide each term in $x y = 3 + 4 x$ by $x$ and simplify.

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Step 3.4.3.1

Divide each term in $x y = 3 + 4 x$ by $x$ .

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

Step 3.4.3.2

Simplify the left side.

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Step 3.4.3.2.1

Cancel the common factor of $x$ .

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Step 3.4.3.2.1.1

Cancel the common factor.

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

Step 3.4.3.2.1.2

Divide $y$ by $1$ .

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

Step 3.4.3.3

Simplify the right side.

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Step 3.4.3.3.1

Cancel the common factor of $x$ .

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Step 3.4.3.3.1.1

Cancel the common factor.

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

Step 3.4.3.3.1.2

Divide $4$ by $1$ .

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

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.2.3.2

Cancel the common factor of $3$ .

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Step 5.2.3.2.1

Cancel the common factor.

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

Step 5.2.3.2.2

Rewrite the expression.

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

Step 5.2.4

Combine the opposite terms in $x - 4 + 4$ .

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Step 5.2.4.1

Add $- 4$ and $4$ .

$f^{- 1} (\frac{3}{x - 4}) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

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

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

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

Step 5.3.3

Simplify the denominator.

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Step 5.3.3.1

Subtract $4$ from $4$ .

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

Step 5.3.3.2

Add $\frac{3}{x}$ and $0$ .

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

Step 5.3.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3.5

Cancel the common factor of $3$ .

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Step 5.3.5.1

Cancel the common factor.

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

Step 5.3.5.2

Rewrite the expression.

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

Step 5.4

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

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