How to Graph Piecewise Functions and Find Limits

TL;DR

To graph a piecewise function, graph each piece separately over its stated condition, then combine them into one graph. In the example f(x) = x² for x < 0 and (3/4)x − 2 for x ≥ 0, the parabola forms the left side and the line forms the right, and the graph jumps from a y value of 0 to −2 at x = 0, a jump discontinuity. This gives a domain of (−∞, ∞) and a range of [−2, ∞). Read on for the limits and end behavior.

Transcript

in this video we're going to focus on graphing peie wise functions identifying the domain and range and we're going to go over limits and continuity as well so let's begin let's say if f ofx is equal to x^2 and 3X over 4 - 2 let's say it equals x² when X is less than zero and it equals 34 x - 2 when X is greater than or equal to zero so how can we ... Read More

Key Insights

• 📈 Graphing a piecewise function involves plotting each part separately and combining them.
• ❣️ The domain of a piecewise function is determined by the allowed x values, while the range is dependent on the corresponding y values.
• 😥 Different types of discontinuities, such as jump discontinuity and point discontinuity, can occur in a piecewise function.
• 👈 Analyzing the limits of a piecewise function helps understand its behavior towards infinity or specific x values. Left and right limits are examined separately.

Questions & Answers

Q: How do you find the domain and range of a piecewise function?

The domain is all allowed x values; in the example x can be anything because there are no fractions with x in the denominator, no radicals, and no logarithms, so the domain is (−∞, ∞). For the range, read the y values from bottom to top: the lowest y value is −2 (included, so a bracket) and the highest goes to positive infinity, giving a range of [−2, ∞).

Q: How do you graph a piecewise function?

Graph each piece separately based on its condition, then combine the pieces into a single graph. In the example, x² provides the left side of a parabola for x < 0, and the linear equation (3/4)x − 2 provides the right side for x ≥ 0. Plot points for each piece and pay attention to open versus closed circles at the endpoints.

Q: When do you use an open circle versus a closed circle at an endpoint?

Use an open circle when the condition excludes the value and a closed circle when it includes it. In the example, x² applies when x < 0 (not equal to 0), so there is an open circle at x = 0, while (3/4)x − 2 applies when x ≥ 0, which includes 0, so that endpoint gets a closed circle.

Q: What type of discontinuity does the example piecewise function have?

It has a jump discontinuity at x = 0. The graph disconnects there and jumps from a y value of 0 to a y value of −2, so you can literally see the jump. This differs from a point discontinuity, which is simply a hole in the graph.

Q: How do you graph the line (3/4)x − 2?

Write it in y = mx + b form: the y-intercept b is −2 and the slope m is 3/4. Start at the y-intercept of −2, then rise 3 units up and run 4 units to the right to reach the point (4, 1). You can confirm this by plugging in x = 4: (3/4)(4) − 2 = 3 − 2 = 1.

Q: How do you determine the limit of a piecewise function at a specific x value?

Evaluate the left-hand limit and the right-hand limit separately as x approaches that value, then compare them. In the example, as x approaches 0 from the left the y value approaches 0, and from the right it approaches −2. Because these two values do not match, the limit as x approaches 0 does not exist, even though each one-sided limit exists.

Q: What is the end behavior of this piecewise function?

End behavior describes the limits as x approaches negative and positive infinity. Here, as x approaches negative infinity (the left end behavior) the blue parabola goes up, so y approaches positive infinity. As x approaches positive infinity (the right end behavior) the line also goes up, so y again approaches positive infinity.

Summary & Key Takeaways

• The video discusses how to graph piecewise functions by first graphing each part separately and then combining them into a single graph.

• It explains how to identify the domain and range of a piecewise function by considering the allowed x values and the corresponding y values.

• The video also covers how to determine the type of discontinuity in a piecewise function, such as point discontinuity or jump discontinuity.

• It explores how to find the limits of a piecewise function and analyze the left end behavior and right end behavior.