Finding The Range Of F(x) = 9x - 17 With Domain {1, 3, 5}
Finding the range of a function is a fundamental concept in mathematics, particularly within the study of functions and their properties. The range represents the set of all possible output values that a function can produce when given its defined domain. In simpler terms, it's the collection of 'y' values you get after plugging in all the allowed 'x' values. For the function f(x) = 9x - 17 with the domain {1, 3, 5}, we will delve into how to determine its range systematically. The domain, in this context, is a discrete set of values, which simplifies our task of finding the range. This exploration is not only crucial for understanding the behavior of functions but also forms the basis for more advanced mathematical concepts. Before diving into the calculations, it's important to understand the nature of linear functions and how their properties affect their range.
Demystifying Functions, Domain, and Range
To accurately determine the range of f(x) = 9x - 17, it's essential to first establish a clear understanding of functions, domains, and ranges. A function in mathematics is a relationship between a set of inputs and a set of permissible outputs, with the defining characteristic that each input is associated with exactly one output. Visualize it as a machine where you input a number (x), and the machine processes it according to a specific rule to produce a unique output (f(x)). The domain of a function is the complete set of all possible input values, the x-values, that the function can accept. In contrast, the range is the complete set of all possible output values, the f(x)-values, that the function produces when given the inputs from its domain. The range is essentially the image of the domain under the function's mapping rule. Understanding this distinction is crucial for accurately calculating the range. In the context of the given function, f(x) = 9x - 17, the rule is a linear equation, and the domain is specified as a discrete set of values. This discrete domain simplifies the process of finding the range, as we only need to evaluate the function at each of the given x-values.
Dissecting the Linear Function f(x) = 9x - 17
The function under scrutiny, f(x) = 9x - 17, is a linear function. Linear functions are characterized by their straight-line graphs and can be generally represented in the form f(x) = mx + b, where m is the slope and b is the y-intercept. In our case, the slope m is 9, indicating that for every unit increase in x, the value of f(x) increases by 9 units. The y-intercept b is -17, which is the point where the line crosses the y-axis. The properties of linear functions make finding their range straightforward, especially when the domain is a finite set of values, as in our problem. A linear function is a polynomial function of degree one or zero, meaning the highest power of the variable x is 1. This simple structure allows us to predict the function's behavior across its domain. The constant slope ensures a consistent rate of change, which is crucial when determining the range for a given domain. Understanding that f(x) = 9x - 17 is a linear function is key to choosing the correct method for finding its range over the specified domain {1, 3, 5}.
Step-by-Step Calculation of the Range for f(x) = 9x - 17
Now, let's methodically compute the range of the function f(x) = 9x - 17 for the domain {1, 3, 5}. To do this, we need to evaluate the function at each value within the domain.
- Evaluate f(1): Substitute x = 1 into the function: f(1) = 9(1) - 17 = 9 - 17 = -8
- Evaluate f(3): Substitute x = 3 into the function: f(3) = 9(3) - 17 = 27 - 17 = 10
- Evaluate f(5): Substitute x = 5 into the function: f(5) = 9(5) - 17 = 45 - 17 = 28
By substituting each x-value from the domain into the function, we obtain the corresponding f(x) values. These f(x) values form the range of the function for the given domain. This step-by-step calculation ensures that we systematically consider each element of the domain and its effect on the output of the function. The simplicity of the linear function f(x) = 9x - 17 makes this evaluation process straightforward and accurate.
Determining the Range: The Set of Output Values
After calculating the function values for each element in the domain, we can now determine the range. From our calculations:
- f(1) = -8
- f(3) = 10
- f(5) = 28
The range of the function f(x) = 9x - 17 for the domain {1, 3, 5} is the set of these output values. Therefore, the range is {-8, 10, 28}. This set encompasses all the possible outputs of the function when the inputs are restricted to the given domain. The range, expressed as a set, provides a concise representation of the function's behavior over the specified domain. Understanding the range allows us to analyze the function's output characteristics and its limitations. In this case, the range consists of three distinct values, corresponding to the three elements in the domain. The range is a key concept in understanding the behavior and characteristics of any function, as it defines the boundaries of its possible outputs.
Range of f(x) = 9x - 17: Final Answer
In conclusion, after systematically evaluating the function f(x) = 9x - 17 for each element in the domain {1, 3, 5}, we have accurately determined the range. By substituting x = 1, x = 3, and x = 5 into the function, we obtained the corresponding output values of -8, 10, and 28, respectively. Therefore, the range of the function f(x) = 9x - 17 for the given domain is {-8, 10, 28}. This final answer encapsulates the complete set of possible output values for the function within the specified domain. The range provides a comprehensive view of the function's behavior and its output limitations. This process of finding the range underscores the fundamental relationship between a function's domain and its corresponding outputs. Understanding and accurately determining the range is a critical skill in the study of functions and their applications in mathematics and other fields.
Visualizing the Function and Its Range
While we have analytically determined the range of f(x) = 9x - 17 for the domain {1, 3, 5}, it's beneficial to visualize this function to further solidify our understanding. Since f(x) = 9x - 17 is a linear function, its graph is a straight line. However, because our domain is discrete, we're not considering the entire line, but rather just three points on it. These points correspond to the coordinates (1, -8), (3, 10), and (5, 28). Plotting these points on a coordinate plane provides a visual representation of the function's behavior over the given domain. This visualization helps to connect the algebraic representation of the function with its graphical interpretation. The y-coordinates of these points visually represent the range of the function, clearly showing the output values corresponding to the input values in the domain. Visualizing the function reinforces the concept of the range as the set of all possible output values. It also aids in understanding how the function transforms input values into output values. A graphical representation is a powerful tool in comprehending functions and their properties, particularly the relationship between the domain and the range.
The Significance of Range in Mathematical Contexts
The concept of the range of a function extends far beyond this specific example. It is a cornerstone in various areas of mathematics, including calculus, analysis, and linear algebra. Understanding the range is crucial for determining the invertibility of a function, as a function must be surjective (onto) to have an inverse. The range also plays a significant role in optimization problems, where we often seek to find the maximum or minimum value of a function within its range. In real-world applications, the range can represent physical limitations, such as the possible temperatures in a chemical reaction or the feasible output levels in an economic model. For example, when modeling population growth, the range would represent the possible population sizes, which must be non-negative values. Similarly, in signal processing, the range of a signal function might represent the amplitude levels, which have practical limitations. The range provides essential information about the function's behavior and its applicability in different contexts. Its significance underscores the importance of thoroughly understanding and accurately determining the range of a function.
Extending the Concept: Beyond Discrete Domains
While we have focused on a function with a discrete domain, it's important to consider how the concept of range extends to functions with continuous domains, such as the set of all real numbers. For functions with continuous domains, the range may be an interval or a union of intervals. Determining the range for these functions often involves analyzing the function's behavior, including its increasing and decreasing intervals, local maxima and minima, and asymptotic behavior. Calculus techniques, such as finding critical points and evaluating limits, are commonly used to determine the range of functions with continuous domains. For example, consider the quadratic function f(x) = x^2. Its domain is all real numbers, but its range is [0, ∞) because the square of any real number is non-negative. Similarly, the range of trigonometric functions like sine and cosine is [-1, 1], regardless of their domain. Understanding how to determine the range for both discrete and continuous domains is crucial for a comprehensive understanding of functions and their properties. The techniques used for continuous domains build upon the fundamental concept of the range established in simpler cases like the discrete domain example we explored.
