Find the Inverse g(x)=1/(x-1)

$g (x) = \frac{1}{x - 1}$

Step 1

Write $g (x) = \frac{1}{x - 1}$ as an equation.

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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

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

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y - 1, 1$

Remove parentheses.

$y - 1, 1$

The LCM of one and any expression is the expression.

$y - 1$

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

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

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

Simplify the left side.

Cancel the common factor of $y - 1$ .

Cancel the common factor.

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

Rewrite the expression.

$1 = x (y - 1)$

Simplify the right side.

Simplify by multiplying through.

Apply the distributive property.

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

Move $- 1$ to the left of $x$ .

$1 = x y - 1 \cdot x$

Rewrite $- 1 x$ as $- x$ .

$1 = x y - x$

Solve the equation.

Rewrite the equation as $x y - x = 1$ .

$x y - x = 1$

Add $x$ to both sides of the equation.

$x y = 1 + x$

Divide each term in $x y = 1 + x$ by $x$ and simplify.

Divide each term in $x y = 1 + x$ by $x$ .

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

Simplify the left side.

Cancel the common factor of $x$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Cancel the common factor of $x$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

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

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

Step 5

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

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

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

Set up the composite result function.

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

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

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

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $x - 1$ by $1$ .

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

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

Add $- 1$ and $1$ .

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

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

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

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

Simplify the denominator.

Subtract $1$ from $1$ .

$g (\frac{1}{x} + 1) = \frac{1}{\frac{1}{x} + 0}$

Add $\frac{1}{x}$ and $0$ .

$g (\frac{1}{x} + 1) = \frac{1}{\frac{1}{x}}$

Multiply the numerator by the reciprocal of the denominator.

$g (\frac{1}{x} + 1) = 1 x$

Multiply $x$ by $1$ .

$g (\frac{1}{x} + 1) = x$

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

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

Find the Inverse g(x)=1/(x-1)

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