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Chapter 1: Problem 26

Also, give the (a) \(x\) -intercept (if any), (b) \(y\) -intercept (if anyy, (c) domain, (d) range, and (e) slope of the line (if defined). Do not use a calculator. \(x=2\)

Short Answer

Expert verified
(a) No x-intercept, (b) No y-intercept, (c) Domain: \(\{2\}\), (d) Range: \(\mathbb{R}\), (e) Slope: Undefined.

Step by step solution

01

Identify the equation type

The equation given is \(x = 2\). This is the equation of a vertical line at \(x = 2\) in the Cartesian plane.
02

Find the x-intercept

The x-intercept of a line occurs where the line crosses the x-axis. In this vertical line equation \(x = 2\), every point on the line has an \(x\)-value of 2. However, it will never cross the x-axis because there is no \(y\)-value related to an intercept point on the x-axis, as it would require \(y = 0\) to intersect the axis exactly at \(x\), which isn't shown as part of the line's definition here.
03

Find the y-intercept

A vertical line like \(x = 2\) does not have a y-intercept. This is because it never crosses the y-axis, given all points of the line have an \(x\)-value of 2, which is offset from being able to touch the y-axis.
04

Determine the domain

The domain of a function refers to all possible \(x\)-values that you can input into the equation. For the line \(x = 2\), the only \(x\)-value it takes is 2. Hence, the domain is \(\{2\}\).
05

Determine the range

The range of a function refers to all possible \(y\)-values that can occur. Because \(x = 2\) represents a vertical line that extends infinitely in both vertical directions, the range will be all real numbers \((\mathbb{R})\).
06

Find the slope of the line

The slope of a line is calculated as \(\frac{\Delta y}{\Delta x}\). For a vertical line like \(x = 2\), the \(\Delta x = 0\), which means division by zero in the slope formula. Therefore, the slope for a vertical line is undefined.

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Key Concepts

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

Intercepts
Intercepts are points where a line crosses the axes on a graph. For most lines, finding these points helps determine how the line interacts with the plane. However, vertical lines like the one described by the equation \( x = 2 \) behave a little differently.
  • x-intercept: This is the point where the line crosses the x-axis. Normally, a line will intersect the x-axis at a point \((x, 0)\), but our vertical line is parallel to the y-axis and will never actually touch the x-axis, meaning it doesn't have an x-intercept.

  • y-intercept: This is where the line would cross the y-axis. For vertical lines such as \( x = 2 \), this never happens since these lines only extend parallel to the y-axis without ever meeting it. Thus, there is no y-intercept for a vertical line.
Domain and Range
When discussing lines, the domain and range refer to the sets of possible inputs and outputs that the line can accept and produce.
  • Domain: For vertical lines like \( x = 2 \), the domain is quite limited. Unlike other lines that have a range of x-values, this vertical line is restricted, with \( x \) always being equal to 2. Thus, the domain is precisely \( \{2\} \), as this is the only x-value the line accepts.

  • Range: The range is the set of y-values that the line can take. Since \( x = 2 \) extends infinitely upwards and downwards, any y-value is possible for this line. Therefore, the range is \((\mathbb{R})\), which includes all real numbers.
Slope of a Line
The slope of a line measures how steep it is and the direction it goes. It's calculated as the change in \( y \) over the change in \( x \) (\( \frac{\Delta y}{\Delta x} \)). For most lines, this is a straightforward computation.
  • However, with a vertical line like \( x = 2 \), a problem arises. The change in \( x \) (\( \Delta x \)) is zero since the line doesn't move left or right. When you try to calculate the slope with \( \Delta x = 0 \), you end up trying to divide by zero.

  • In mathematics, division by zero is undefined. Therefore, the slope of a vertical line is considered undefined. Essentially, a vertical line is so steep that it doesn't have a measurable slope in the traditional sense.

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Most popular questions from this chapter

Rainfall By noon, 3 inches of rain had fallen during a storm. Rain continued to fall at a rate of \(\frac{1}{4}\) inch per hour. (a) Find a formula for a linear function \(f\) that models the amount of rainfall \(x\) hours past noon. (b) Find the total amount of rainfall by 2: 30 P.M.

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