Piecewise Functions Explained Examples And Graphs

Hey guys! Today, we're diving deep into the fascinating world of piecewise functions. Piecewise functions, those mathematical chameleons, are defined by different formulas across various intervals of their domain. They're like a set of instructions, each applying only to a specific input range. In this article, we'll dissect a specific piecewise function, explore its behavior, and understand how each piece contributes to the overall function.

Unveiling the Piecewise Function f(x)

Let's start by examining the piecewise function we'll be working with:

$$f(x)=\left\{\begin{array}{ll} -\frac{1}{4} x^2+6 x+36, & x < -2 \\ 4 x-15, & -2 \leq x < 4 \\ 3^{x-4}, & x > 4 \end{array}\right.$$

This function, f(x), isn't defined by a single equation. Instead, it's composed of three distinct pieces, each with its own formula and domain. Think of it as three mini-functions stitched together to create a larger, more complex one. Let's break down each piece:

• Piece 1: For values of x less than -2 (x < -2), the function is defined as f(x) = -1/4 x^2 + 6x + 36. This is a quadratic function, meaning it will form a parabola when graphed. The -1/4 coefficient in front of the x² term indicates that the parabola opens downwards. Understanding this quadratic behavior is crucial for grasping the function's characteristics in this interval. We'll explore how to find the vertex and other key features of this parabola later on.
• Piece 2: When x is between -2 (inclusive) and 4 (-2 ≤ x < 4), the function follows the rule f(x) = 4x - 15. This is a linear function, so it will produce a straight line. The slope of this line is 4, meaning it rises steeply as x increases. Linear functions are straightforward to analyze, making this piece relatively simple to understand. The linear segment provides a contrasting behavior compared to the quadratic piece.
• Piece 3: Finally, for x greater than 4 (x > 4), the function is defined as f(x) = 3^(x-4). This is an exponential function, which means its value grows rapidly as x increases. The base of the exponent is 3, indicating a growth rate faster than, say, 2^(x-4). Exponential functions are known for their rapid growth, and this piece will significantly impact the function's behavior for large values of x.

Each piece of this function plays a crucial role in defining its overall behavior. Understanding the individual characteristics of each piece – the quadratic, the linear, and the exponential – is the first step in mastering piecewise functions. We'll delve deeper into each piece, exploring their graphs, key features, and how they connect to form the complete function.

Graphing the Piecewise Function

To truly understand a piecewise function, visualizing its graph is invaluable. The graph allows us to see how the different pieces connect and how the function behaves across its entire domain. So, let's embark on graphing f(x), considering each piece separately and then combining them. This graphical representation will solidify our understanding.

1. Graphing the Quadratic Piece: For x < -2, we have f(x) = -1/4 x² + 6x + 36. This is a downward-opening parabola. To graph it accurately, we need to find its vertex and a few points. The x-coordinate of the vertex can be found using the formula x = -b / 2a, where a = -1/4 and b = 6. Plugging in these values, we get x = -6 / (2 * -1/4) = 12. However, this vertex lies outside our defined domain (x < -2). So, we need to consider the endpoint x = -2. When x = -2, f(x) = -1/4 (-2)² + 6(-2) + 36 = -1 - 12 + 36 = 23. This gives us the point (-2, 23). Since the parabola opens downwards, we can choose a few more points to the left of x = -2 to sketch the curve. For example, when x = -4, f(x) = -1/4 (-4)² + 6(-4) + 36 = -4 - 24 + 36 = 8, giving us the point (-4, 8). Similarly, when x = -6, f(x) = -1/4 (-6)² + 6(-6) + 36 = -9 - 36 + 36 = -9, giving us the point (-6, -9). We plot these points and sketch the parabolic curve for x < -2. Remember to use an open circle at (-2, 23) since x = -2 is not included in this piece's domain. Sketching this parabola is the first step in visualizing the function.
2. Graphing the Linear Piece: For -2 ≤ x < 4, we have f(x) = 4x - 15. This is a straight line. We can find two points on this line to graph it. At x = -2, f(x) = 4(-2) - 15 = -8 - 15 = -23, giving us the point (-2, -23). At x = 4, f(x) = 4(4) - 15 = 16 - 15 = 1, giving us the point (4, 1). We plot these two points and draw a straight line connecting them. Use a closed circle at (-2, -23) since x = -2 is included in this piece's domain, and an open circle at (4, 1) since x = 4 is not. This linear segment is a crucial part of the function.
3. Graphing the Exponential Piece: For x > 4, we have f(x) = 3^(x-4). This is an exponential function with a base of 3. At x = 4, f(x) = 3^(4-4) = 3^0 = 1. However, since x > 4, we use an open circle at (4, 1). As x increases, the function grows rapidly. For example, at x = 5, f(x) = 3^(5-4) = 3^1 = 3, giving us the point (5, 3). At x = 6, f(x) = 3^(6-4) = 3^2 = 9, giving us the point (6, 9). We plot these points and sketch the exponential curve for x > 4. The exponential curve shows the rapid growth of the function.

By combining these three pieces, we get the complete graph of the piecewise function f(x). The graph will show a parabola segment for x < -2, a line segment for -2 ≤ x < 4, and an exponential curve for x > 4. Pay close attention to the endpoints and whether they are included (closed circles) or excluded (open circles) in each piece. The complete graph provides a holistic view of the function's behavior.

Analyzing Key Features and Behavior

Once we have the graph, we can analyze the key features and behavior of the piecewise function. This includes identifying the domain, range, intervals of increasing and decreasing, and any points of discontinuity. Analyzing these features provides a deeper understanding of the function.

• Domain: The domain of f(x) is all real numbers because each piece covers a portion of the number line. The quadratic piece covers x < -2, the linear piece covers -2 ≤ x < 4, and the exponential piece covers x > 4. Together, these intervals cover the entire real number line. The domain is a fundamental characteristic of the function.
• Range: The range is a bit more complex. The quadratic piece, for x < -2*, reaches a maximum value (though not at its vertex). The linear piece spans a range of values, and the exponential piece extends to infinity. By looking at the graph, we can determine the range to be y ≥ -23. The range provides insights into the possible output values.
• Increasing and Decreasing Intervals: The quadratic piece is decreasing for x < -2. The linear piece is increasing. The exponential piece is also increasing. Thus, we can identify the intervals where the function's value goes up or down. Monotonicity helps understand the function's trend.
• Discontinuity: A crucial aspect of piecewise functions is the possibility of discontinuities. Discontinuities occur where the pieces don't connect smoothly. In our function, there's a discontinuity at x = -2, where the quadratic and linear pieces meet, and another discontinuity at x = 4, where the linear and exponential pieces meet. The function