91Ó°ÊÓ

When graphing functions, especially piecewise functions, understanding each segment is crucial. A piecewise function is defined by multiple expressions, each valid on different intervals of the domain. Let's look at our function, which has two segments.
For the part where \(x \leq -1\), we have the function \(f(x) = 3x - 1\). This is a linear equation with a slope of 3, meaning it rises quickly. It crosses the y-axis at -1, meaning it cuts through the point \((0, -1)\) if extended. But, since we only consider \(x \leq -1\), the graph starts there and moves leftward.
For \(x > -1\), the function changes to \(f(x) = x\), another simple line with a slope of 1. This line moves diagonally upward through the origin. However, starting our graph at \(x = -1\), it extends to the right. Note that these lines don't connect, leading to a break or discontinuity at \(x = -1\). This break is represented in the graph by an open circle at \(x = -1\) on this line, indicating it's not included.

Continuity is the property of a function to be smooth and unbroken. A function is considered continuous if you can draw its graph without lifting your pen from the paper. For a piecewise function, like the one given, assessing continuity requires checking at points where the function's rule changes.
In our example, check continuity at \(x = -1\), where the function jumps from \(f(x) = 3x - 1\) to \(f(x) = x\). For each interval, \((-\infty, -1]\) and \((-1, +\infty)\), the function is continuous; the lines are smooth with no interruptions. However, at \(x = -1\), the output from \(f(x) = 3x - 1\) is \(-4\), while \(f(x) = x\) would be \(-1\) if you evaluated directly. The functions do not "match up" at \(x = -1\), causing continuity to break there.

Discontinuities occur where a function is not smooth, that is, where you can't traverse from one segment of the graph to another without jumping or being interrupted. In our piecewise function, the discontinuity happens at the crossover point \(x = -1\). Here, the piecewise rules switch from \(f(x) = 3x - 1\) to \(f(x) = x\), creating a situation where the two pieces of the function do not meet at the same point on the y-axis.
This kind of discontinuity is known as a jump discontinuity since there's a sudden shift in the value from \(-4\) to \(-1\) as you move across \(x = -1\). Such discontinuities divide the graph into separate, unconnected parts. When analyzing functions, identifying these points is vital for understanding how the function behaves across its domain and for more advanced calculus concepts which may build upon this understanding, such as finding limits.