Find the Inverse F(x)=2x-1

$F (x) = 2 x - 1$

Write $F (x) = 2 x - 1$ as an equation.

$y = 2 x - 1$

Interchange the variables.

$x = 2 y - 1$

Solve for $y$ .

Rewrite the equation as $2 y - 1 = x$ .

$2 y - 1 = x$

Add $1$ to both sides of the equation.

$2 y = x + 1$

Divide each term in $2 y = x + 1$ by $2$ and simplify.

Divide each term in $2 y = x + 1$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

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

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

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

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

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

Set up the composite result function.

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

Evaluate $F^{- 1} (2 x - 1)$ by substituting in the value of $F$ into $F^{- 1}$ .

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

Combine the numerators over the common denominator.

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

Combine the opposite terms in $2 x - 1 + 1$ .

Add $- 1$ and $1$ .

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

Add $2 x$ and $0$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

$F^{- 1} (2 x - 1) = x$

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

Set up the composite result function.

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

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

$F (\frac{x}{2} + \frac{1}{2}) = 2 (\frac{x}{2} + \frac{1}{2}) - 1$

Simplify each term.

Apply the distributive property.

$F (\frac{x}{2} + \frac{1}{2}) = 2 (\frac{x}{2}) + 2 (\frac{1}{2}) - 1$

Cancel the common factor of $2$ .

Cancel the common factor.

$F (\frac{x}{2} + \frac{1}{2}) = 2 (\frac{x}{2}) + 2 (\frac{1}{2}) - 1$

Rewrite the expression.

$F (\frac{x}{2} + \frac{1}{2}) = x + 2 (\frac{1}{2}) - 1$

Cancel the common factor of $2$ .

Cancel the common factor.

$F (\frac{x}{2} + \frac{1}{2}) = x + 2 (\frac{1}{2}) - 1$

Rewrite the expression.

$F (\frac{x}{2} + \frac{1}{2}) = x + 1 - 1$

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

Subtract $1$ from $1$ .

$F (\frac{x}{2} + \frac{1}{2}) = x + 0$

Add $x$ and $0$ .

$F (\frac{x}{2} + \frac{1}{2}) = x$

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

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

Find the Inverse F(x)=2x-1

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