Precalculus Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

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

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

Step 3.2

Find the LCD of the terms in the equation.

Tap for more steps...

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.

$y, 1$

Step 3.2.2

The LCM of one and any expression is the expression.

$y$

Step 3.3

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

Tap for more steps...

Step 3.3.1

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

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

Step 3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.1

Cancel the common factor of $y$ .

Tap for more steps...

Step 3.3.2.1.1

Cancel the common factor.

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

Step 3.3.2.1.2

Rewrite the expression.

$y + 7 = x y$

Step 3.4

Solve the equation.

Tap for more steps...

Step 3.4.1

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

$y + 7 - x y = 0$

Step 3.4.2

Subtract $7$ from both sides of the equation.

$y - x y = - 7$

Step 3.4.3

Factor $y$ out of $y - x y$ .

Tap for more steps...

Step 3.4.3.1

Factor $y$ out of $y^{1}$ .

$y \cdot 1 - x y = - 7$

Step 3.4.3.2

Factor $y$ out of $- x y$ .

$y \cdot 1 + y (- x) = - 7$

Step 3.4.3.3

Factor $y$ out of $y \cdot 1 + y (- x)$ .

$y (1 - x) = - 7$

Step 3.4.4

Divide each term in $y (1 - x) = - 7$ by $1 - x$ and simplify.

Tap for more steps...

Step 3.4.4.1

Divide each term in $y (1 - x) = - 7$ by $1 - x$ .

$\frac{y (1 - x)}{1 - x} = \frac{- 7}{1 - x}$

Step 3.4.4.2

Simplify the left side.

Tap for more steps...

Step 3.4.4.2.1

Cancel the common factor of $1 - x$ .

Tap for more steps...

Step 3.4.4.2.1.1

Cancel the common factor.

$\frac{y (1 - x)}{1 - x} = \frac{- 7}{1 - x}$

Step 3.4.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 7}{1 - x}$

Step 3.4.4.3

Simplify the right side.

Tap for more steps...

Step 3.4.4.3.1

Move the negative in front of the fraction.

$y = - \frac{7}{1 - x}$

Step 4

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

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

Step 5

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

Tap for more steps...

Step 5.1

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

Step 5.2

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

Tap for more steps...

Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

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

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

Step 5.2.3

Simplify the denominator.

Tap for more steps...

Step 5.2.3.1

Write $1$ as a fraction with a common denominator.

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

Step 5.2.3.2

Combine the numerators over the common denominator.

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

Step 5.2.3.3

Rewrite $\frac{x - (x + 7)}{x}$ in a factored form.

Tap for more steps...

Step 5.2.3.3.1

Apply the distributive property.

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

Step 5.2.3.3.2

Multiply $- 1$ by $7$ .

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

Step 5.2.3.3.3

Subtract $x$ from $x$ .

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

Step 5.2.3.3.4

Subtract $7$ from $0$ .

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

Step 5.2.3.4

Move the negative in front of the fraction.

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

Step 5.2.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.2.5

Cancel the common factor of $7$ .

Tap for more steps...

Step 5.2.5.1

Move the leading negative in $- \frac{x}{7}$ into the numerator.

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

Step 5.2.5.2

Cancel the common factor.

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

Step 5.2.5.3

Rewrite the expression.

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

Step 5.3

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

Tap for more steps...

Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

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

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

Step 5.3.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3.4

To write $7$ as a fraction with a common denominator, multiply by $\frac{1 - x}{1 - x}$ .

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

Step 5.3.5

Simplify terms.

Tap for more steps...

Step 5.3.5.1

Combine the numerators over the common denominator.

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

Step 5.3.5.2

Rewrite using the commutative property of multiplication.

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

Step 5.3.5.3

Cancel the common factor of $1 - x$ .

Tap for more steps...

Step 5.3.5.3.1

Move the leading negative in $- \frac{- 7 + 7 (1 - x)}{1 - x}$ into the numerator.

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

Step 5.3.5.3.2

Cancel the common factor.

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

Step 5.3.5.3.3

Rewrite the expression.

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

Step 5.3.6

Simplify each term.

Tap for more steps...

Step 5.3.6.1

Apply the distributive property.

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

Step 5.3.6.2

Multiply $7$ by $1$ .

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

Step 5.3.6.3

Multiply $- 1$ by $7$ .

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

Step 5.3.7

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 5.3.7.1

Add $- 7$ and $7$ .

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

Step 5.3.7.2

Subtract $7 x$ from $0$ .

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

Step 5.3.7.3

Cancel the common factor of $7$ .

Tap for more steps...

Step 5.3.7.3.1

Factor $7$ out of $- (- 7 x)$ .

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

Step 5.3.7.3.2

Cancel the common factor.

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

Step 5.3.7.3.3

Rewrite the expression.

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

Step 5.3.7.4

Multiply.

Tap for more steps...

Step 5.3.7.4.1

Multiply $- 1$ by $- 1$ .

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

Step 5.3.7.4.2

Multiply $x$ by $1$ .

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

Step 5.4

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

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