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Chapter 9: Problem 61

Graph each linear or constant function. Give the domain and range. \(g(x)=-4\)

Short Answer

Expert verified
Domain: (-鈭�, 鈭�), Range: {-4}, Horizontal line at y = -4.

Step by step solution

01

Title - Identify the Function Type

The function given is a constant function, as it does not depend on the variable x. Instead, it always equals -4.
02

Title - Understand the Graph

A constant function like this one is represented by a horizontal line on the graph. In this case, the line is horizontal and runs at y = -4.
03

Title - Draw the Graph

Draw a horizontal line that goes through the point (0, -4) on the y-axis. This line should extend infinitely in both the positive and negative directions along the x-axis.
04

Title - Determine the Domain

The domain of a constant function is all real numbers, written as \(\text{Domain: } (-\infty, \infty)\). This means the function is defined for any value of x.
05

Title - Determine the Range

The range of a constant function is just the one value that the function takes, written as \(\text{Range: } \{-4\}\). This means the function output is always -4, regardless of the input.

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Key Concepts

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

Graphing Linear Functions
Graphing linear functions is an essential skill in algebra and precalculus. A linear function can be written in the form of \(y = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept. This function creates a straight line when graphed.
  • Identify the slope \(m\) and y-intercept \(b\).
  • Plot the y-intercept on the y-axis.
  • Use the slope to find another point on the line by moving up/down and right/left from the y-intercept.
  • Draw a straight line through the points.

Remember that the graph of a constant function is a special case of a linear function, where the slope \(m = 0\), resulting in a horizontal line.
Domain and Range
When dealing with functions, understanding the domain and range is crucial. The domain represents all possible input values (x-values) for the function, while the range signifies all possible output values (y-values).
  • The domain of most linear functions is all real numbers, \((-\infty, \infty)\), since you can input any real number into the function.
  • The range depends on the function's type.

For a constant function like \(g(x) = -4\):
  • The domain is \((-\infty, \infty)\), all real numbers can be input.
  • The range is \(\{-4\}\), as the function only outputs the value -4.

Understanding domain and range helps in visualizing and analyzing functions, making it easier to predict and graph their behavior.
Horizontal Line
A horizontal line is a straight line where all points share the same y-value, resulting from a constant function. For example, in the function \(g(x) = -4\):
  • This indicates the function value is always -4, regardless of x.
  • The horizontal line is positioned at y = -4 on the y-axis.

To graph a horizontal line:
  • Identify the y-value, in this case, -4.
  • Draw a straight line through the y-value point on the y-axis, extending infinitely in both positive and negative directions along the x-axis.

This simplicity makes horizontal lines easy to graph, as they do not change with x, providing a clear visual representation of constant functions.

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Most popular questions from this chapter

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