Precalculus Examples

$f (x) = \frac{5}{2 - x}$

Step 1

Write $f (x) = \frac{5}{2 - x}$ as an equation.

$y = \frac{5}{2 - x}$

Step 2

Interchange the variables.

$x = \frac{5}{2 - y}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{5}{2 - y} = x$ .

$\frac{5}{2 - y} = x$

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.

$2 - y, 1$

Step 3.2.2

Remove parentheses.

$2 - y, 1$

Step 3.2.3

The LCM of one and any expression is the expression.

$2 - y$

Step 3.3

Multiply each term in $\frac{5}{2 - y} = x$ by $2 - y$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $\frac{5}{2 - y} = x$ by $2 - y$ .

$\frac{5}{2 - y} (2 - y) = x (2 - y)$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $2 - y$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{5}{2 - y} (2 - y) = x (2 - y)$

Step 3.3.2.1.2

Rewrite the expression.

$5 = x (2 - y)$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Apply the distributive property.

$5 = x \cdot 2 + x (- y)$

Step 3.3.3.2

Reorder.

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Step 3.3.3.2.1

Move $2$ to the left of $x$ .

$5 = 2 \cdot x + x (- y)$

Step 3.3.3.2.2

Rewrite using the commutative property of multiplication.

$5 = 2 x - x y$

Step 3.4

Solve the equation.

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Step 3.4.1

Rewrite the equation as $2 x - x y = 5$ .

$2 x - x y = 5$

Step 3.4.2

Subtract $2 x$ from both sides of the equation.

$- x y = 5 - 2 x$

Step 3.4.3

Divide each term in $- x y = 5 - 2 x$ by $- x$ and simplify.

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Step 3.4.3.1

Divide each term in $- x y = 5 - 2 x$ by $- x$ .

$\frac{- x y}{- x} = \frac{5}{- x} + \frac{- 2 x}{- x}$

Step 3.4.3.2

Simplify the left side.

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Step 3.4.3.2.1

Dividing two negative values results in a positive value.

$\frac{x y}{x} = \frac{5}{- x} + \frac{- 2 x}{- x}$

Step 3.4.3.2.2

Cancel the common factor of $x$ .

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Step 3.4.3.2.2.1

Cancel the common factor.

$\frac{x y}{x} = \frac{5}{- x} + \frac{- 2 x}{- x}$

Step 3.4.3.2.2.2

Divide $y$ by $1$ .

$y = \frac{5}{- x} + \frac{- 2 x}{- x}$

Step 3.4.3.3

Simplify the right side.

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Step 3.4.3.3.1

Simplify each term.

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Step 3.4.3.3.1.1

Move the negative in front of the fraction.

$y = - \frac{5}{x} + \frac{- 2 x}{- x}$

Step 3.4.3.3.1.2

Cancel the common factor of $x$ .

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Step 3.4.3.3.1.2.1

Cancel the common factor.

$y = - \frac{5}{x} + \frac{- 2 x}{- x}$

Step 3.4.3.3.1.2.2

Rewrite the expression.

$y = - \frac{5}{x} + \frac{- 2}{- 1}$

Step 3.4.3.3.1.2.3

Move the negative one from the denominator of $\frac{- 2}{- 1}$ .

$y = - \frac{5}{x} - 1 \cdot - 2$

Step 3.4.3.3.1.3

Rewrite $- 1 \cdot - 2$ as $- - 2$ .

$y = - \frac{5}{x} - - 2$

Step 3.4.3.3.1.4

Multiply $- 1$ by $- 2$ .

$y = - \frac{5}{x} + 2$

Step 4

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

$f^{- 1} (x) = - \frac{5}{x} + 2$

Step 5

Verify if $f^{- 1} (x) = - \frac{5}{x} + 2$ is the inverse of $f (x) = \frac{5}{2 - x}$ .

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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 $f^{- 1} (\frac{5}{2 - x})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{5}{2 - x}) = - \frac{5}{\frac{5}{2 - x}} + 2$

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.

$f^{- 1} (\frac{5}{2 - x}) = - (5 (\frac{2 - x}{5})) + 2$

Step 5.2.3.2

Cancel the common factor of $5$ .

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Step 5.2.3.2.1

Cancel the common factor.

$f^{- 1} (\frac{5}{2 - x}) = - (5 (\frac{2 - x}{5})) + 2$

Step 5.2.3.2.2

Rewrite the expression.

$f^{- 1} (\frac{5}{2 - x}) = - (2 - x) + 2$

Step 5.2.3.3

Apply the distributive property.

$f^{- 1} (\frac{5}{2 - x}) = - 1 \cdot 2 + x + 2$

Step 5.2.3.4

Multiply $- 1$ by $2$ .

$f^{- 1} (\frac{5}{2 - x}) = - 2 + x + 2$

Step 5.2.3.5

Multiply $- - x$ .

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Step 5.2.3.5.1

Multiply $- 1$ by $- 1$ .

$f^{- 1} (\frac{5}{2 - x}) = - 2 + 1 x + 2$

Step 5.2.3.5.2

Multiply $x$ by $1$ .

$f^{- 1} (\frac{5}{2 - x}) = - 2 + x + 2$

Step 5.2.4

Combine the opposite terms in $- 2 + x + 2$ .

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Step 5.2.4.1

Add $- 2$ and $2$ .

$f^{- 1} (\frac{5}{2 - x}) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

$f^{- 1} (\frac{5}{2 - x}) = x$

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 $f (- \frac{5}{x} + 2)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{5}{x} + 2) = \frac{5}{2 - (- \frac{5}{x} + 2)}$

Step 5.3.3

Simplify the denominator.

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Step 5.3.3.1

Apply the distributive property.

$f (- \frac{5}{x} + 2) = \frac{5}{2 + \frac{5}{x} - 1 \cdot 2}$

Step 5.3.3.2

Multiply $- - \frac{5}{x}$ .

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Step 5.3.3.2.1

Multiply $- 1$ by $- 1$ .

$f (- \frac{5}{x} + 2) = \frac{5}{2 + 1 (\frac{5}{x}) - 1 \cdot 2}$

Step 5.3.3.2.2

Multiply $\frac{5}{x}$ by $1$ .

$f (- \frac{5}{x} + 2) = \frac{5}{2 + \frac{5}{x} - 1 \cdot 2}$

Step 5.3.3.3

Multiply $- 1$ by $2$ .

$f (- \frac{5}{x} + 2) = \frac{5}{2 + \frac{5}{x} - 2}$

Step 5.3.3.4

Subtract $2$ from $2$ .

$f (- \frac{5}{x} + 2) = \frac{5}{\frac{5}{x} + 0}$

Step 5.3.3.5

Add $\frac{5}{x}$ and $0$ .

$f (- \frac{5}{x} + 2) = \frac{5}{\frac{5}{x}}$

Step 5.3.4

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{5}{x} + 2) = 5 (\frac{x}{5})$

Step 5.3.5

Cancel the common factor of $5$ .

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Step 5.3.5.1

Cancel the common factor.

$f (- \frac{5}{x} + 2) = 5 (\frac{x}{5})$

Step 5.3.5.2

Rewrite the expression.

$f (- \frac{5}{x} + 2) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{5}{x} + 2$ is the inverse of $f (x) = \frac{5}{2 - x}$ .

$f^{- 1} (x) = - \frac{5}{x} + 2$