find f^−1(y) such that f(x)=1+(x)/2-(x)

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!

Learn how to find the inverse function for f(x) = 1 + \frac{x}{2} - x with a detailed step-by-step solution. This problem covers key algebraic concepts, including inverse functions and simplification techniques. Suitable for students in Grades 9-12.

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