Part 1: Domain and Range

Domain: The domain consists of all $x$-values where the graph is defined. Based on the description, the graph starts at $x = -2$ and ends at $x = 4$, including both endpoints.

Thus, the domain is: $\boxed{[-2, 4]}$

Range: The range consists of all $y$-values where the graph is defined. From the description, the lowest $y$-value is $0$ (at $x = -2$ and $x = 2$) and the highest $y$-value is $4$ (at $x = 0$ and $x = 4$).

Thus, the range is: $\boxed{[0, 4]}$

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Part 2: Evaluate the function

Using the graph to find the corresponding function values:

• (a) $f(-2)$: From the graph, at $x = -2$, $y = 0$. $f(-2) = \boxed{0}$

• (b) $f(0)$: From the graph, at $x = 0$, $y = 4$. $f(0) = \boxed{4}$

• (c) $f\left(\frac{1}{2}\right)$: Since $x = \frac{1}{2}$ lies on the line segment between $x = 0$ and $x = 2$, we can interpolate. The line segment falls from $(0, 4)$ to $(2, 0)$. The slope is: $\text{slope} = \frac{0 - 4}{2 - 0} = -2$ So, for $x = \frac{1}{2}$: $f\left(\frac{1}{2}\right) = 4 + (-2) \times \frac{1}{2} = 4 - 1 = \boxed{3}$

• (d) Values of $x$ such that $f(x) = 2$: From the graph, $f(x) = 2$ occurs at two points:

  • On the increasing segment from $(-2, 0)$ to $(0, 4)$, it occurs when $x = -1$.
  • On the decreasing segment from $(0, 4)$ to $(2, 0)$, it occurs when $x = 1$.

Thus, $f(x) = 2$ at $x = \boxed{-1 \text{ and } 1}$.

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Let me know if you need any further explanation!

Related Questions:

1. How do we calculate the slope of a line segment between two points?
2. How can we use interpolation to estimate values on a line segment?
3. What is the difference between the domain and the range of a function?
4. How can we identify the values of $x$ for a given $f(x)$?
5. What is the significance of closed points in a graph?

Tip: When analyzing piecewise functions, always break down the function into its segments to properly evaluate each part!