Algebra Examples

$y = \frac{- 4}{x} + 1$

Step 1

Interchange the variables.

$x = \frac{- 4}{y} + 1$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{- 4}{y} + 1 = x$ .

$\frac{- 4}{y} + 1 = x$

Step 2.2

Subtract $1$ from both sides of the equation.

$\frac{- 4}{y} = x - 1$

Step 2.3

Move the negative in front of the fraction.

$- \frac{4}{y} = x - 1$

Step 2.4

Find the LCD of the terms in the equation.

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Step 2.4.1

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 2.4.2

The LCM of one and any expression is the expression.

$y$

Step 2.5

Multiply each term in $- \frac{4}{y} = x - 1$ by $y$ to eliminate the fractions.

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Step 2.5.1

Multiply each term in $- \frac{4}{y} = x - 1$ by $y$ .

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

Step 2.5.2

Simplify the left side.

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Step 2.5.2.1

Cancel the common factor of $y$ .

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Step 2.5.2.1.1

Move the leading negative in $- \frac{4}{y}$ into the numerator.

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

Step 2.5.2.1.2

Cancel the common factor.

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

Step 2.5.2.1.3

Rewrite the expression.

$- 4 = x y - y$

Step 2.6

Solve the equation.

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Step 2.6.1

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

$x y - y = - 4$

Step 2.6.2

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

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Step 2.6.2.1

Factor $y$ out of $x y$ .

$y x - y = - 4$

Step 2.6.2.2

Factor $y$ out of $- y$ .

$y x + y \cdot - 1 = - 4$

Step 2.6.2.3

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

$y (x - 1) = - 4$

Step 2.6.3

Divide each term in $y (x - 1) = - 4$ by $x - 1$ and simplify.

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Step 2.6.3.1

Divide each term in $y (x - 1) = - 4$ by $x - 1$ .

$\frac{y (x - 1)}{x - 1} = \frac{- 4}{x - 1}$

Step 2.6.3.2

Simplify the left side.

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Step 2.6.3.2.1

Cancel the common factor of $x - 1$ .

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Step 2.6.3.2.1.1

Cancel the common factor.

$\frac{y (x - 1)}{x - 1} = \frac{- 4}{x - 1}$

Step 2.6.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 4}{x - 1}$

Step 2.6.3.3

Simplify the right side.

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Step 2.6.3.3.1

Move the negative in front of the fraction.

$y = - \frac{4}{x - 1}$

Step 3

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

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

Step 4

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

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

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

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

Step 4.2.3

Simplify the denominator.

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Step 4.2.3.1

Subtract $1$ from $1$ .

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

Step 4.2.3.2

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

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

Step 4.2.3.3

Move the negative in front of the fraction.

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

Step 4.2.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2.5

Cancel the common factor of $4$ .

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Step 4.2.5.1

Move the leading negative in $- \frac{x}{4}$ into the numerator.

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

Step 4.2.5.2

Cancel the common factor.

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

Step 4.2.5.3

Rewrite the expression.

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

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

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

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

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3.3.2

Cancel the common factor of $4$ .

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Step 4.3.3.2.1

Move the leading negative in $- \frac{x - 1}{4}$ into the numerator.

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

Step 4.3.3.2.2

Factor $4$ out of $- 4$ .

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

Step 4.3.3.2.3

Cancel the common factor.

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

Step 4.3.3.2.4

Rewrite the expression.

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

Step 4.3.3.3

Multiply $- 1$ by $- 1$ .

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

Step 4.3.3.4

Multiply $x - 1$ by $1$ .

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

Step 4.3.4

Combine the opposite terms in $x - 1 + 1$ .

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Step 4.3.4.1

Add $- 1$ and $1$ .

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

Step 4.3.4.2

Add $x$ and $0$ .

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

Step 4.4

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

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