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Chapter 2: Problem 19

Graph each line. Give the domain and range. $$x=3$$

Short Answer

Expert verified
Domain: {3}; Range: (-鈭�, 鈭�)

Step by step solution

01

Understand the Equation

The equation given is a vertical line. For any vertical line, the equation is in the form of \( x = a \) where \( a \) is a constant.
02

Plot the Line on a Graph

To plot the line \( x = 3 \), find the point on the x-axis where \( x = 3 \). From this point, draw a straight vertical line.
03

Determine the Domain

The domain of a vertical line \( x = 3 \) includes only one value: \( x = 3 \). Hence, the domain is \[ \{3\} \].
04

Determine the Range

The range of a vertical line is all possible values of \( y \). Since the line extends infinitely up and down, the range is \[ (-\infty, \infty) \].

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Key Concepts

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

vertical line equation
A vertical line has a unique property: it runs straight up and down. This means it does not slant or angle but is perfectly perpendicular to the x-axis. The equation for any vertical line is in the form of \( x = a \), where \( a \) is a constant value. This equation tells us that no matter what point you choose on the line, the x-coordinate will always be \( a \). For example, in the exercise \( x = 3 \), this means every point on the line will have an x-coordinate of 3. Understanding this is key to recognizing and plotting vertical lines on a graph.
domain of vertical line
The domain of a function is all the possible values that x can take. For a vertical line like \( x = 3 \), x is always 3. This means the line is fixed in place at \( x = 3 \) and does not extend horizontally. As a result, the domain is very limited and consists of only one value: \( x = 3 \). Therefore, we can write the domain as \({3}\). This limitation is a unique characteristic of vertical lines and distinguishes them from other types of lines which usually have a broader domain.
range of vertical line
While the domain of a vertical line is constrained, its range is quite the opposite. The range of a function includes all possible values that y can take. For a vertical line like \( x = 3 \), there are no restrictions on the values of y. The line extends infinitely upwards and downwards, which means that it covers all possible y-values. Therefore, the range of a vertical line is \((-\infty, \infty)\). This means the line reaches in both positive and negative directions along the y-axis without any bounds.

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