Finding The Range Of F(x) = -2x + 7 For Domain [-2, 3, 8]
Introduction to Range of a Function
In mathematics, understanding the behavior of functions is crucial. One essential aspect of function analysis is determining its range. The range of a function is the set of all possible output values (y-values) that the function can produce for a given set of input values (x-values), known as the domain. In simpler terms, if you feed a function all the values from its domain, the range is the collection of all the results you get back. Determining the range is vital for understanding the function's limitations and its overall behavior. For a linear function like the one we're about to explore, f(x) = -2x + 7, the range is closely related to its domain and the function's slope. This article will meticulously walk you through the process of finding the range of this specific linear function over a given domain, highlighting the key concepts and steps involved. We will delve into how the linear nature of the function simplifies the process, and how understanding the endpoints of the domain helps in efficiently determining the range. Through clear explanations and step-by-step calculations, you'll gain a solid understanding of how to find the range of a function, which is a fundamental skill in algebra and calculus.
Understanding the Function f(x) = -2x + 7
Before we dive into finding the range, let's first dissect the function itself. The function f(x) = -2x + 7 is a linear function, meaning its graph is a straight line. Linear functions have a general form of f(x) = mx + b, where m represents the slope and b represents the y-intercept. In our case, m = -2 and b = 7. The slope, -2, tells us that for every increase of 1 in the x-value, the y-value decreases by 2. This negative slope indicates that the function is decreasing; as x increases, f(x) decreases. The y-intercept, 7, is the point where the line crosses the y-axis, which occurs when x = 0. Understanding the slope and y-intercept is crucial for visualizing and analyzing the function's behavior. A negative slope means that the function's output will decrease as the input increases. This is a key observation that will help us determine the range over the given domain. We will see how the decreasing nature of the function influences the range, especially when we consider the endpoints of the domain. By understanding these fundamental aspects of linear functions, we can approach the problem of finding the range with greater confidence and clarity. The interplay between the slope, y-intercept, and domain will dictate the possible output values of the function.
Defining the Domain: [-2, 3, 8]
The domain of a function is the set of all possible input values (x-values) for which the function is defined. In this case, the domain is given as [-2, 3, 8]. This notation implies that we are considering only these three specific x-values: -2, 3, and 8. It's important to note that this is a discrete domain, meaning we are dealing with a finite set of values rather than a continuous interval. This distinction is crucial because it simplifies the process of finding the range. With a discrete domain, we only need to evaluate the function at each of the given x-values. Unlike continuous domains where we might need to consider the function's behavior over an interval, here we have a limited number of points to work with. Understanding the domain is the first step in determining the range. The domain sets the boundaries for our input values, and consequently, it influences the possible output values. The specific values in the domain will directly determine the values we calculate for the range. This discrete nature of the domain allows us to directly compute the corresponding function values, which will then form the range of the function for this specific domain.
Calculating the Function Values for Each Domain Element
Now that we understand the function f(x) = -2x + 7 and its domain [-2, 3, 8], we can proceed to calculate the function values for each element in the domain. This involves substituting each x-value from the domain into the function and evaluating the expression. Let's start with x = -2:
f(-2) = -2(-2) + 7 = 4 + 7 = 11
Next, we evaluate the function for x = 3:
f(3) = -2(3) + 7 = -6 + 7 = 1
Finally, we calculate the function value for x = 8:
f(8) = -2(8) + 7 = -16 + 7 = -9
These calculations give us the corresponding y-values (or function values) for each x-value in the domain. We have found that f(-2) = 11, f(3) = 1, and f(8) = -9. These values are the outputs of the function for the given inputs, and they are the building blocks for determining the range. The process of substituting each x-value and evaluating the function is straightforward but essential. It directly translates the input values into their corresponding output values, providing us with the set of numbers that constitute the range. This step is crucial in understanding how the function transforms the domain into the range.
Determining the Range from the Calculated Values
After calculating the function values for each element in the domain, we have the following results:
f(-2) = 11f(3) = 1f(8) = -9
The range of the function for the given domain is the set of these output values. Therefore, the range of f(x) = -2x + 7 for the domain [-2, 3, 8] is {11, 1, -9}. The range is simply the collection of all the y-values that we obtained by plugging in the x-values from the domain. In this case, the range consists of three distinct numbers. It's important to present the range as a set, which is why we use curly braces {}. The order of the elements in the set doesn't matter, but it's common practice to list them in either ascending or descending order for clarity. In this instance, we have listed them in descending order, showcasing the highest value first and the lowest value last. This step solidifies our understanding of the function's output over the specified domain. By collecting and presenting these output values as a set, we accurately define the range of the function for the given domain.
Conclusion: The Range of f(x) = -2x + 7 for Domain [-2, 3, 8]
In conclusion, we have successfully determined the range of the function f(x) = -2x + 7 for the domain [-2, 3, 8]. By understanding the nature of linear functions, calculating the function values for each element in the domain, and collecting these values into a set, we found that the range is {11, 1, -9}. This process highlights the fundamental relationship between the domain and range of a function. The domain acts as the input set, while the range represents the corresponding output set generated by the function. For this specific linear function, the negative slope played a crucial role in determining the range, as it indicated a decreasing function. This meant that the largest x-value in the domain corresponded to the smallest y-value in the range, and vice versa. Understanding these concepts is essential for analyzing functions and their behavior. The ability to find the range is a valuable skill in mathematics, applicable in various contexts, from solving equations to modeling real-world phenomena. By following the steps outlined in this article, you can confidently find the range of similar functions over discrete domains. The methodical approach of evaluating the function at each domain element provides a clear and concise way to determine the set of all possible output values, thus defining the range accurately.
