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Chapter 10: Problem 26

Find the domain and range of the function. $$g:\\{(-4,4),(3,8),(4,5),(9,-2),(10,-7)\\}$$

Short Answer

Expert verified
The domain of the function \(g\) is \(-4,3,4,9,10\) and the range is \(4,8,5,-2,-7\)

Step by step solution

01

Identify the Set of Ordered Pairs

The given function is defined as a set of ordered pairs, which look like this: \( g=\{(-4,4),(3,8),(4,5),(9,-2),(10,-7)\} \). Each ordered pair represents a point on the graph of the function where the first number is the x-value and the second number is the y-value.
02

Identify the Domain

To find the domain, list all the x-values from each ordered pair. The order of the values is not important, but each value should only appear once. Here, the x-values for each pair are -4, 3, 4, 9 and 10.Therefore, the domain of the function g is \(-4,3,4,9,10\)
03

Identify the Range

Similarly, to find the range of the function, list all the unique y-values from each ordered pair. Here, the y-values for each pair are 4, 8, 5, -2, and -7. Therefore, the range of the function g is \(4,8,5,-2,-7\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ordered Pairs
When we talk about functions in mathematics, we describe them using ordered pairs. An ordered pair is simply a pair of numbers used to locate a point on a graph, with the first number representing the horizontal position, and the second the vertical position. The general notation for an ordered pair is \( (x, y) \), where \( x \) is the x-value and \( y \) is the y-value.

In the context of the exercise, the function \( g \) is defined by a set of such pairs, listing the specific points at which the function takes certain values. Understanding ordered pairs is crucial as they are the building blocks for plotting the graph of the function and identifying its domain and range.
Graph of a Function
A graph of a function visually displays the relationship between x-values and y-values. It consists of points in a coordinate system where each point corresponds to an ordered pair from the function.

The graph can provide a comprehensive picture of the function's behavior, showing where it increases, decreases, and where it may have constant values. It's an essential tool for understanding the nature of the function beyond the numerical data. However, in cases where we only have a set of ordered pairs without an explicit rule for the function, such as in our exercise, the graph would consist of discrete points.
X-Values
The term x-values refers to the set of all possible first elements in the ordered pairs of a function. X-values are also known as the 'input' or 'independent' values in a function and are found along the horizontal axis of the graph. They are a vital component in determining the domain of a function, which is the complete set of possible x-values.

For our given function \( g \) with the ordered pairs \( \{(-4,4),(3,8),(4,5),(9,-2),(10,-7)\} \), the x-values are identified in the first position of each pair, showing the distinct inputs over which the function is defined.
Y-Values
In contrast to x-values, y-values are the second elements in ordered pairs and represent the 'output' or 'dependent' values in a function. They are plotted along the vertical axis of a graph. The collection of y-values from the function鈥檚 ordered pairs defines the range of the function, which details all the possible outcomes or outputs.

In our exercise's context, the y-values from the given function \( g \) are identified as the second elements in the ordered pairs and are critical for determining the set of values the function can produce.

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