Graphing And Range Of Piecewise Function F(x) = {1/2x^2, X < 2; 4x-3, X ≥ 2}

July 11, 2025 by Scholario Team 77 views

Graphing and Determining the Range of a Piecewise Function

In the fascinating world of mathematics, piecewise functions hold a special place. They are functions defined by multiple sub-functions, each applying to a specific interval of the domain. This unique structure allows piecewise functions to model complex relationships and behaviors that single functions cannot capture. In this article, we will explore a specific piecewise function, delve into its properties, and visualize its graph. We will also learn how to determine the range of a piecewise function using its graph. Specifically, we will consider the piecewise function:

f(x)=\left\{\begin{array}{cc}\frac{1}{2} x^2 & x<2 \\ 4 x-3 & x \geq 2\end{array}\right.

Our goal is to understand its domain, graph it accurately, and then use the graph to identify its range. This process will illustrate the power and versatility of piecewise functions in mathematical modeling.

Understanding the Domain of the Piecewise Function

In this section, we'll delve into understanding the domain of the piecewise function given by:

$f(x)=\left\{\begin{array}{cc}\frac{1}{2} x^2 & x<2 \\ 4 x-3 & x \geq 2\end{array}\right.$

The domain of a function is the set of all possible input values (x-values) for which the function is defined. In simpler terms, it's the range of x-values that you can plug into the function and get a valid output. For piecewise functions, determining the domain involves examining the intervals over which each sub-function is defined. Piecewise functions, by their very nature, are defined in 'pieces,' each with its own specific interval. These intervals collectively contribute to the overall domain of the function.

For our function, we have two sub-functions:

$\frac{1}{2}x^2$ which is defined for $x < 2$
$4x - 3$ which is defined for $x \geq 2$

The first sub-function, $\frac{1}{2}x^2$ , is a quadratic function, specifically a parabola. This part of the piecewise function is defined for all values of x strictly less than 2. This means that any number smaller than 2 can be inputted into this part of the function, and we will get a corresponding output. There is no lower limit to the x-values in this interval, so it extends to negative infinity.

The second sub-function, $4x - 3$ , is a linear function, forming a straight line when graphed. This part of the function is defined for all values of x greater than or equal to 2. This means that 2 itself and any number larger than 2 can be inputted into this part of the function. This interval extends from 2 (inclusive) to positive infinity.

Combining these two intervals, we can see that the function is defined for all real numbers. The first part covers all numbers less than 2, and the second part covers 2 and all numbers greater than 2. There is no gap or undefined point in the x-values. Therefore, the domain of the piecewise function is the set of all real numbers, which can be expressed in interval notation as $(-\infty, \infty)$ .

Graphing the Piecewise Function

Now, let's move on to graphing the piecewise function. Graphing a piecewise function involves plotting each sub-function over its respective interval. This requires careful attention to the endpoints of the intervals and the behavior of each sub-function.

To graph our piecewise function:

$f(x)=\left\{\begin{array}{cc}\frac{1}{2} x^2 & x<2 \\ 4 x-3 & x \geq 2\end{array}\right.$

We will follow a step-by-step approach:

Graph the first sub-function, $\frac{1}{2}x^2$ , for $x < 2$ :
- This is a parabola. To graph it, we can choose some x-values less than 2 and calculate the corresponding f(x) values.
- Let's take x = -2, -1, 0, 1. The corresponding f(x) values are:
  - x = -2, f(x) = $\frac{1}{2}(-2)^2 = 2$
  - x = -1, f(x) = $\frac{1}{2}(-1)^2 = 0.5$
  - x = 0, f(x) = $\frac{1}{2}(0)^2 = 0$
  - x = 1, f(x) = $\frac{1}{2}(1)^2 = 0.5$
- Plot these points. Since the interval is $x < 2$ , we also need to consider the point at x = 2, but we'll use an open circle at this point to indicate that it's not included in this part of the function.
- At x = 2, f(x) = $\frac{1}{2}(2)^2 = 2$ . So, we'll place an open circle at (2, 2).
- Draw a smooth curve connecting the points to form the parabola for $x < 2$ .
Graph the second sub-function, $4x - 3$ , for $x \geq 2$ :
- This is a linear function, so we need at least two points to draw the line.
- Since the interval is $x \geq 2$ , let's start with x = 2.
- At x = 2, f(x) = $4(2) - 3 = 5$ . So, we'll place a closed circle at (2, 5) since this point is included in this part of the function.
- Let's take another point, x = 3. At x = 3, f(x) = $4(3) - 3 = 9$ .
- Plot the point (3, 9).
- Draw a straight line passing through (2, 5) and (3, 9) for $x \geq 2$ .
Combine the two graphs:
- The graph of the piecewise function will consist of the parabola for $x < 2$ and the line for $x \geq 2$ .
- Pay attention to the point where the two sub-functions meet at x = 2. The open circle at (2, 2) from the parabola indicates that this point is not included in the first part, while the closed circle at (2, 5) from the line indicates that this point is included in the second part.

By following these steps, you will obtain an accurate graph of the piecewise function. This graph visually represents the function's behavior over its entire domain.

Determining the Range of the Piecewise Function

Having graphed the piecewise function, we now turn our attention to determining its range. The range of a function is the set of all possible output values (f(x) values or y-values) that the function can produce. In other words, it's the set of all y-values that the graph of the function covers.

To determine the range of our piecewise function:

$f(x)=\left\{\begin{array}{cc}\frac{1}{2} x^2 & x<2 \\ 4 x-3 & x \geq 2\end{array}\right.$

We will analyze the graph we constructed in the previous section:

Consider the first sub-function, $\frac{1}{2}x^2$ , for $x < 2$ :
- This part of the function is a parabola that opens upwards. We graphed it for x values less than 2.
- The lowest point of this parabola occurs at the vertex, which is at (0, 0). This means that the minimum y-value for this part of the function is 0.
- As x approaches 2 from the left, f(x) approaches $\frac{1}{2}(2)^2 = 2$ . However, since x is strictly less than 2, this point (2, 2) is not included in this part of the function (indicated by an open circle on the graph).
- Therefore, the y-values for this part of the function range from 0 (inclusive) up to 2 (exclusive). In interval notation, this is [0, 2).
Consider the second sub-function, $4x - 3$ , for $x \geq 2$ :
- This part of the function is a straight line with a positive slope. We graphed it for x values greater than or equal to 2.
- At x = 2, f(x) = $4(2) - 3 = 5$ . This is the starting y-value for this part of the function, and it is included (indicated by a closed circle on the graph).
- As x increases, f(x) also increases without bound. This means that the y-values for this part of the function extend to positive infinity.
- Therefore, the y-values for this part of the function range from 5 (inclusive) to infinity. In interval notation, this is [5, $\infty$ ).
Combine the ranges of the two sub-functions:
- The overall range of the piecewise function is the union of the ranges of its sub-functions.
- We have [0, 2) from the first part and [5, $\infty$ ) from the second part.
- Since these intervals do not overlap, the combined range is [0, 2) $\cup$ [5, $\infty$ ).

Thus, the range of the piecewise function is the set of all y-values greater than or equal to 0 but less than 2, as well as all y-values greater than or equal to 5.

Conclusion

In this comprehensive exploration, we have successfully analyzed a piecewise function, determined its domain, graphed it accurately, and identified its range. We began by understanding that the domain of the given piecewise function, $f(x)=\left\{\begin{array}{cc}\frac{1}{2} x^2 & x<2 \\ 4 x-3 & x \geq 2\end{array}\right.$ , is $(-\infty, \infty)$ , meaning it is defined for all real numbers. This understanding laid the groundwork for further analysis.

Next, we graphed the function by plotting each sub-function over its specified interval. This involved graphing a parabola for $x < 2$ and a straight line for $x \geq 2$ . The graph provided a visual representation of the function's behavior and was crucial for determining its range.

Finally, by carefully examining the graph, we determined the range of the piecewise function to be [0, 2) $\cup$ [5, $\infty$ ). This means the function's output values include all numbers from 0 up to (but not including) 2, as well as all numbers from 5 to infinity. This result highlights how piecewise functions can have complex ranges due to their segmented nature.

This exercise demonstrates the importance of understanding the properties of different types of functions and how they combine in piecewise functions. It also showcases the power of graphical analysis in determining key characteristics of functions, such as their range. Piecewise functions are versatile tools in mathematics and are used to model various real-world phenomena, making their understanding essential for any mathematics enthusiast or student. The ability to graph and analyze these functions is a valuable skill in mathematical problem-solving and modeling.