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 $x = 1$).