Graphing & Analyzing A Piecewise Function: A Step-by-Step Guide

Hey guys! Today, we're diving into the fascinating world of piecewise functions. Specifically, we're going to graph the following piecewise function and then determine its domain and range. Trust me, it's not as intimidating as it sounds! Let's break it down together.

Understanding Piecewise Functions

Before we jump into graphing, let's quickly recap what a piecewise function actually is. A piecewise function is a function defined by multiple sub-functions, each applying to a certain interval of the main function's domain. Think of it like a set of different rules that apply depending on the input value, x. It's essential to pay close attention to the intervals and their corresponding functions to accurately graph and analyze the whole thing.

In our example, we have:

$$f(x) = \begin{cases} x+2 & \text{if } x < -1 \\ 3 & \text{if } x \geq -1 \end{cases}$$

This means that for any x value less than -1, we use the function x + 2. For any x value greater than or equal to -1, we simply use the constant function 3. That's it! Now, let's get our hands dirty with graphing.

Part A: Graphing the Piecewise Function

To graph this piecewise function, we'll graph each piece separately over its specified interval. We'll use the graphing tools to illustrate this. I'll walk you through the process, step-by-step.

Step 1: Graphing $f(x) = x + 2$ for $x < -1$

First, consider the linear function $f(x) = x + 2$. This is a straight line with a slope of 1 and a y-intercept of 2. However, we only want to graph this line for x values less than -1. This means our graph will start at x = -1, but not include that point. We use an open circle at x = -1 to indicate this.

To find the y-value at x = -1, plug -1 into the equation: f(-1) = -1 + 2 = 1. So, we'll have an open circle at the point (-1, 1). Now, let's pick another point where x < -1, such as x = -2. Then, f(-2) = -2 + 2 = 0. This gives us the point (-2, 0). We can now draw a line starting from the open circle at (-1, 1) and passing through (-2, 0), extending to the left (since x < -1).

Important Note: Always remember that open circle! It's crucial because it tells us that the function approaches but does not include that point.

Step 2: Graphing $f(x) = 3$ for $x \geq -1$

Next, we consider the constant function $f(x) = 3$ for $x \geq -1$. This means that for all x values greater than or equal to -1, the function's value is always 3. Graphically, this is a horizontal line at y = 3.

Since this piece is defined for $x \geq -1$, we include the point at x = -1. Thus, we use a closed circle (or a solid dot) at the point (-1, 3). From this point, we simply draw a horizontal line extending to the right, since all x values greater than -1 are included.

Combining the Pieces

When you combine both pieces, you'll see that the graph consists of a straight line (with an open circle at (-1, 1)) extending to the left, and a horizontal line (with a closed circle at (-1, 3)) extending to the right. The jump at x = -1 is a key characteristic of this piecewise function.

Part B: Determining the Domain and Range

Now that we've successfully graphed our piecewise function, let's figure out its domain and range. Understanding these two concepts is fundamental in analyzing any function.

Domain

The domain of a function is the set of all possible input values (x values) for which the function is defined. In other words, it's all the x values that you can plug into the function without causing any mathematical errors (like dividing by zero or taking the square root of a negative number).

For our piecewise function, let's examine each piece:

• The first piece, $f(x) = x + 2$, is defined for all $x < -1$. This interval includes all real numbers less than -1.
• The second piece, $f(x) = 3$, is defined for all $x \geq -1$. This interval includes all real numbers greater than or equal to -1.

Combining these two intervals, we see that the function is defined for all real numbers. There are no breaks or gaps in the x values that are covered by either piece. Therefore, the domain of our piecewise function is all real numbers.

In interval notation, we write this as: $(-\infty, \infty)$.

Range

The range of a function is the set of all possible output values (y values) that the function can produce. It's the set of all values that $f(x)$ can take.

Again, let's look at each piece:

• For $f(x) = x + 2$ where $x < -1$, the function can take any value less than 1 (but not equal to 1, due to the open circle at (-1, 1)).
• For $f(x) = 3$ where $x \geq -1$, the function only takes the value 3.

So, the range consists of all real numbers less than 1, plus the single value 3. We can express this as the interval $(-\infty, 1)$ combined with the single point {3}.

In set notation, we write this as: $(-\infty, 1) \cup \{3\}$.

Putting It All Together

So, to recap:

• Piecewise Function: $f(x) = \begin{cases} x+2 & \text{if } x < -1 \\ 3 & \text{if } x \geq -1 \end{cases}$
• Domain: $(-\infty, \infty)$
• Range: $(-\infty, 1) \cup \{3\}$

And that's it! We've successfully graphed the piecewise function and determined its domain and range. Piecewise functions might seem complicated at first, but with a bit of practice, you'll get the hang of them in no time.

Keep practicing, and don't be afraid to ask questions. You got this!