Let's start by simplifying the given function $f(x)$:

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

First, combine the terms involving $x$:

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

Since $x = \frac{2x}{2}$, we can rewrite it as:

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

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

Now that we have the simplified function:

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

Finding the Inverse

To find the inverse function $f^{-1}(y)$, we will follow these steps:

1. Start by setting $y = f(x)$:

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

2. Solve for $x$ in terms of $y$:

   Subtract 1 from both sides: $y - 1 = -\frac{x}{2}$

   Multiply both sides by -2 to solve for $x$: $x = -2(y - 1)$

   Simplifying: $x = -2y + 2$

Thus, the inverse function is:

$f^{-1}(y) = -2y + 2$

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Do you have any questions or would you like further details?

Related Questions:

1. How do you find the domain and range of inverse functions?
2. Can a function have more than one inverse?
3. How can we verify if $f^{-1}(y)$ is correct by composition of functions?
4. What are the applications of inverse functions in real life?
5. How can you graph the inverse function $f^{-1}(y)$ from $f(x)$?

Tip: When finding an inverse function, always check by composing the original and inverse functions to verify correctness!