https://file.aiquickdraw.com/mathbot/file/17260208457745vkolm2u.jpg

The problem asks to find the **domain** and **range** of the piecewise function:

\[
f(x) = 
\begin{cases}
4x - 10, & \text{if } x \leq 2, \\
4 - \frac{1}{2}x, & \text{if } x > 2.
\end{cases}
\]

### 1. Domain:
The domain is the set of all possible values of \(x\) for which the function is defined. Both pieces of the function are defined for all real numbers, and since one piece is defined for \(x \leq 2\) and the other for \(x > 2\), the function is defined for all real numbers.

So, the domain is:
\[
\boxed{(-\infty, \infty)}
\]

### 2. Range:
To find the range, we analyze the output values for each piece of the function.

- For the first piece \( f(x) = 4x - 10 \) for \(x \leq 2\):
  - The value at \(x = 2\) is \( f(2) = 4(2) - 10 = -2 \).
  - As \(x\) decreases, \(4x - 10\) decreases without bound (since \(4x\) becomes more negative as \(x\) decreases). Thus, the values of \(f(x)\) go to \(-\infty\) as \(x\) approaches \(-\infty\).

  So, the range for this piece is \( (-\infty, -2] \).

- For the second piece \( f(x) = 4 - \frac{1}{2}x \) for \(x > 2\):
  - As \(x\) increases, \(4 - \frac{1}{2}x\) decreases (since the term \(\frac{1}{2}x\) grows larger and subtracts more from 4).
  - At \(x = 2\), \( f(2) = 4 - \frac{1}{2}(2) = 4 - 1 = 3 \).
  - As \(x \to \infty\), the function approaches \(-\infty\).

  So, the range for this piece is \( (-\infty, 3) \).

### Combining both ranges:
The overall range is the union of the two ranges:
\[
\boxed{(-\infty, 3)}
\]

Let me know if you'd like further details or have any questions! Here are some related questions to expand on:

1. How can we determine if a function is continuous at \(x = 2\) for this piecewise function?
2. What is the derivative of each piece of the function, and how does it behave?
3. How can you graph piecewise functions like this one?
4. Are there any points where this function is not differentiable?
5. How does one find the inverse of a piecewise function?

**Tip**: When working with piecewise functions, always carefully check continuity and differentiability at the boundaries where the pieces meet!

Find the domain and range for the function f(x) = {4x - 10, if x <= 2; 4 - (1/2)x, if x > 2}.

Learn how to find the domain and range of a piecewise function: f(x) = {4x - 10, if x <= 2; 4 - (1/2)x, if x > 2}. This problem involves analyzing linear equations in a piecewise-defined context and determining their corresponding ranges.

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