# Question #3a38f

Apr 2, 2016

${f}^{- 1} \left(x\right) = 2 x - 1$

#### Explanation:

${f}^{- 1} \left(x\right)$ is the inverse function of $f \left(x\right)$. All that means is an input for $f \left(x\right)$ equals an output for ${f}^{- 1} \left(x\right)$. For example, the functions $f \left(x\right) = - x$ and ${f}^{- 1} \left(x\right) = x$ are inverses. $f \left(1\right) = - 1$ and ${f}^{- 1} \left(- 1\right) = 1$. An inverse function generates the input for the original function.

You can find inverses in 4 simple steps.

Step 1: Change to $x$ and $y$ Notation
All this means is replace $f \left(x\right)$ with $y$:
$y = \frac{x + 1}{2}$

Step 2: Swap $x$ and $y$
$x$ becomes $y$ and $y$ becomes $x$:
$x = \frac{y + 1}{2}$

Step 3: Solve for $y$
We have $x = \frac{y + 1}{2}$. Solving for $y$ is just a matter of algebra:
$x = \frac{y + 1}{2}$
$2 x = y + 1$
$2 x - 1 = y$

Step 4: Replace $y$ with ${f}^{- 1} \left(x\right)$
We change back to $f \left(x\right)$ notation in this step, adding a $- 1$ to say it's an inverse:
${f}^{- 1} \left(x\right) = 2 x - 1$