Question

State the domain and the range of each function.$$\begin{array}{c|ccccc}\hline x & 2 & 4 & 6 & 8 & 10 \\\\\hline y & 0 & -2 & -5 & -9 & -15 \\\\\hline\end{array}$$

Short answer

The domain of the function is {2, 4, 6, 8, 10} and the range is {0, -2, -5, -9, -15}.

Step-by-step solution

Identify the x-values

Review the given table and list all the x-values, which represent the input of the function.

Identify the y-values

List all the y-values given in the table, as these represent the output of the function.

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.

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 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.

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.

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.