Here is the graph of the piecewise function. The first part is the parabola $f(x) = \frac{1}{4}x^2$ for $x < 1$, and the second part is the linear function $f(x) = 3x - 1$ for $x \geq 1$.

Part b: Determine the range

From the graph, we can observe the following:

• The first part of the function, $f(x) = \frac{1}{4}x^2$, is a parabola that opens upwards with a minimum value at $x = 0$, where $f(0) = 0$. As $x$ decreases, the value of $f(x)$ also decreases without bound, so the range for $x < 1$ is $(-\infty, 0]$.

• The second part, $f(x) = 3x - 1$, is a straight line with slope 3 and a y-intercept of -1. It starts at $f(1) = 2$ and increases indefinitely. Therefore, the range for $x \geq 1$ is $[2, \infty)$.

Combining both parts, the range of the function is:

$(-\infty, 0] \cup [2, \infty)$

Would you like further details or have any questions about this solution?

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Here are some related questions to consider:

1. What happens to the behavior of the piecewise function as $x$ approaches 1 from the left and right?
2. How would the range change if the function's definition for $x \geq 1$ were changed to something like $f(x) = 2x + 1$?
3. How can you identify the domain and range of a piecewise function by analyzing its graph?
4. What effect would changing the coefficient of $x^2$ in the first part of the piecewise function have on the graph?
5. How can you find the limit of a piecewise function as $x$ approaches a particular value, like 1 in this case?

Tip: To graph piecewise functions accurately, break them into intervals based on the conditions, and plot each segment separately!