Finding The Range Of F(x) = X^2 + 5x For Domain [-1, 0, 1]
Understanding the range of a function is a fundamental concept in mathematics. It involves determining all possible output values that the function can produce for a given set of input values. In this article, we will delve into the specifics of finding the range of the function f(x) = x^2 + 5x when its domain is restricted to the set [-1, 0, 1]. This exploration will involve evaluating the function at each point in the domain and identifying the set of corresponding output values. By understanding how the function behaves over this particular domain, we can gain valuable insights into its overall characteristics and behavior. This process not only enhances our comprehension of the function itself but also strengthens our broader understanding of mathematical functions and their properties.
The range of a function represents the set of all possible output values (y-values) that the function can produce when given input values from its domain (x-values). To determine the range of f(x) = x^2 + 5x over the domain [-1, 0, 1], we need to evaluate the function at each point within the domain. This means substituting each value from the domain into the function and calculating the corresponding output. The resulting set of outputs will form the range of the function for the specified domain. This process highlights the direct relationship between the input and output values of a function, providing a clear picture of how the function transforms the input into the output within the given domain. By systematically evaluating the function at each point, we can accurately map the domain to the range, revealing the function's behavior and the spectrum of values it can produce.
When determining the range for the given domain [-1, 0, 1], we must consider each value separately. For x = -1, we substitute -1 into the function: f(-1) = (-1)^2 + 5(-1) = 1 - 5 = -4. This calculation shows that when the input is -1, the function outputs -4. Next, we evaluate the function at x = 0: f(0) = (0)^2 + 5(0) = 0. This result indicates that when the input is 0, the output is also 0. Finally, we substitute x = 1: f(1) = (1)^2 + 5(1) = 1 + 5 = 6. This calculation reveals that when the input is 1, the output is 6. By evaluating the function at each point in the domain, we have identified the specific output values that correspond to those inputs. This step-by-step evaluation is crucial for accurately determining the range of the function over the given domain. Understanding how each input maps to its corresponding output is essential for grasping the function's behavior and its overall range.
Evaluating the Function at Domain Points
To find the range, we begin by substituting each value from the domain [-1, 0, 1] into the function f(x) = x^2 + 5x. This process will give us the corresponding output values for each input, which will then allow us to determine the overall range of the function for the specified domain. The range is the set of all possible output values, so by evaluating the function at each point in the domain, we can identify all the values that the function can take on. This is a fundamental step in understanding the behavior of the function and how it maps input values to output values. The act of substitution is not merely a mechanical process; it is a crucial step in revealing the function's characteristics and its range over a given domain. Through this methodical evaluation, we gain a clearer picture of the function's capabilities and limitations.
Calculation for x = -1
Substituting x = -1 into the function f(x) = x^2 + 5x, we get f(-1) = (-1)^2 + 5(-1). This calculation involves squaring -1, which results in 1, and multiplying 5 by -1, which gives -5. Therefore, the expression becomes f(-1) = 1 - 5. Completing the subtraction, we find that f(-1) = -4. This result indicates that when the input is -1, the function outputs -4. This single calculation is a crucial step in mapping the domain value -1 to its corresponding range value -4. It exemplifies the function's transformation of a specific input into its associated output. Such calculations are fundamental to understanding how the function behaves over a given domain and to identifying the overall range of possible output values.
Calculation for x = 0
Next, we substitute x = 0 into the function f(x) = x^2 + 5x. This yields f(0) = (0)^2 + 5(0). Squaring 0 results in 0, and multiplying 5 by 0 also results in 0. Therefore, the expression simplifies to f(0) = 0 + 0. Adding these values together, we find that f(0) = 0. This result is particularly significant because it demonstrates that when the input is 0, the function outputs 0. The point (0, 0) is a key point on the graph of the function, often indicating an intercept or a critical point. Understanding this relationship between the input and output when x = 0 provides valuable insight into the function's behavior and its position relative to the axes.
Calculation for x = 1
Finally, we substitute x = 1 into the function f(x) = x^2 + 5x, resulting in f(1) = (1)^2 + 5(1). Squaring 1 gives 1, and multiplying 5 by 1 gives 5. The expression then becomes f(1) = 1 + 5. Adding these values, we find that f(1) = 6. This calculation shows that when the input is 1, the function outputs 6. This is another crucial data point that helps us understand the function's behavior and how it transforms input values into output values. The value f(1) = 6 contributes to the overall range of the function within the specified domain, and it is a critical piece of information for sketching the graph of the function or for further analysis.
Determining the Range
After evaluating the function at each point in the domain [-1, 0, 1], we have found the corresponding output values: f(-1) = -4, f(0) = 0, and f(1) = 6. The range of the function f(x) = x^2 + 5x for the given domain is the set of these output values. The range, therefore, consists of the set {-4, 0, 6}. This set represents all possible output values that the function can produce when the input is restricted to the domain [-1, 0, 1]. The process of determining the range involves not only evaluating the function at each point but also collecting and presenting these values as a set, which clearly defines the function's output capabilities over the specified domain. This understanding of the range is essential for various applications, including graphing the function, solving equations, and analyzing the function's behavior.
In conclusion, the range of the function f(x) = x^2 + 5x for the domain [-1, 0, 1] is {-4, 0, 6}. This result was obtained by evaluating the function at each point in the domain and collecting the resulting output values. This exercise illustrates the fundamental process of determining the range of a function for a given domain, a crucial concept in mathematics. Understanding the range is vital for comprehending the behavior of functions and their applications in various mathematical contexts. The methodical approach of substituting domain values and calculating corresponding range values provides a clear and effective way to analyze functions and their output capabilities.
