https://file.aiquickdraw.com/mathbot/file/1726600792393tl3hghnc.jpg

The graph above represents the piecewise function:

- The **blue curve** for \( x < 0 \) is \( f(x) = x^2 + 1 \), a parabola shifted upward.
- The **green line** for \( 0 \leq x \leq 1 \) is \( f(x) = x \), a straight line.
- The **red curve** for \( x > 1 \) is \( f(x) = \sqrt{x} \), a square root function.

### Key Points:
- The domain of the function is \( (-\infty, \infty) \).
- The range of the function is \( [0, \infty) \), since the minimum value \( f(x) \) can take is 0 and the function increases without bound.
- The function is **many-one** because multiple values of \( x \) can map to the same \( f(x) \) value in different intervals (for example, \( f(0) = f(1) = 1 \)).

### Let me know if you have any further questions or need more details!

Here are five related questions for further exploration:
1. How would the range change if the square root function was \( \sqrt{x} + 2 \)?
2. What would be the effect of changing \( f(x) = x^2 + 1 \) to \( f(x) = x^2 - 1 \) for \( x < 0 \)?
3. How would you modify the function to make it **one-one** instead of many-one?
4. Can this function be classified as injective or surjective? Why or why not?
5. What is the significance of the function being continuous or discontinuous at the points \( x = 0 \) and \( x = 1 \)?

**Tip**: When solving piecewise functions, always ensure that the function is well-defined at the boundaries between intervals (like \( x = 0 \) and \( x = 1 \)).

Define function and many-one function. Sketch the graph of the following function and obtain the domain and range of that function.

Learn how to define and graph a piecewise function and determine the domain and range. This problem covers concepts such as many-one functions and piecewise function graphs, with examples like f(x) = x^2 + 1, f(x) = x, and f(x) = √x. Suitable for students in Grades 9-12.

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