Problem 46
Question
State the domain and the range of each function.$$\begin{array}{c|ccccc}\hline x & 2 & 4 & 6 & 8 & 10 \\\\\hline y & 0 & -2 & -5 & -9 & -15 \\\\\hline\end{array}$$
Step-by-Step Solution
Verified Answer
The domain of the function is {2, 4, 6, 8, 10} and the range is {0, -2, -5, -9, -15}.
1Step 1: Identify the x-values
Review the given table and list all the x-values, which represent the input of the function.
2Step 2: Identify the y-values
List all the y-values given in the table, as these represent the output of the function.
3Step 3: Determine the Domain
The domain of a function is the set of all possible input values (x-values) which for this function are the values given in the table.
4Step 4: Determine the Range
The range of a function is the set of all possible output values (y-values) for this function, which are also given in the table.
Key Concepts
Function Input and OutputX-values and Y-values in FunctionsIdentifying Domain and Range
Function Input and Output
Understanding the input and output of a function is like knowing the beginning and end of a journey. In mathematics, a function is a special relationship where each input (x-value) has a single output (y-value). Imagine you are using a vending machine: you select a button (input), and you get a snack (output).
Functions work similarly, taking an 'x' and producing a 'y', often visualized as \( y = f(x) \). For instance, if we input 2 into our function from the exercise, our output is 0, denoted as \( f(2) = 0 \). Grasping this concept can make mathematics much less intimidating, serving as a foundational understanding for more complex topics.
Functions work similarly, taking an 'x' and producing a 'y', often visualized as \( y = f(x) \). For instance, if we input 2 into our function from the exercise, our output is 0, denoted as \( f(2) = 0 \). Grasping this concept can make mathematics much less intimidating, serving as a foundational understanding for more complex topics.
X-values and Y-values in Functions
In the realm of functions, x-values and y-values are the coordinates that make up the function's operation. Think of them as the 'where' in our journey of inputs and outputs. The x-values, which are often independent, represent where we start; each value corresponds to one and only one y-value, which depends on our function's rule.
From our exercise, list the x-values as the starting points: 2, 4, 6, 8, and 10. The related y-values are the destinations of these points: 0, -2, -5, -9, and -15 respectively. These pairs form the foundation of plotting points on a graph, allowing us to visualize the function's behavior.
From our exercise, list the x-values as the starting points: 2, 4, 6, 8, and 10. The related y-values are the destinations of these points: 0, -2, -5, -9, and -15 respectively. These pairs form the foundation of plotting points on a graph, allowing us to visualize the function's behavior.
Identifying Domain and Range
Discovering the domain and range of a function is akin to mapping the limits of its journey. The domain is all the x-values that you can put into a function. In our exercise, these are the ones listed in the table. It's like having a set path to follow.
The range, on the other hand, is all the potential y-values that come out from those x-values, it's the assortment of destinations we can reach following our path. In our exercise, this includes all the y-values listed in the table. Understanding the concept of domain and range is crucial for navigating through functions and knowing the extent of inputs and outputs possible.
The range, on the other hand, is all the potential y-values that come out from those x-values, it's the assortment of destinations we can reach following our path. In our exercise, this includes all the y-values listed in the table. Understanding the concept of domain and range is crucial for navigating through functions and knowing the extent of inputs and outputs possible.
Other exercises in this chapter
Problem 43
Find the inverse of $$y=3(x-2)-4(x+3)$$
View solution Problem 44
State the domain and the range of each function.$$(0,2),(1,4),(2,8),(3,16),(4,32)$$
View solution Problem 47
Find the domain and the range for each function.$$y=\sqrt{x-7}$$
View solution Problem 48
Find the domain and the range for each function.$$y=\frac{3}{\sqrt{x-2}}$$
View solution