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Chapter 2: Problem 22

Graph each line. Give the domain and range. $$-3 x+6=0$$

Short Answer

Expert verified
The line is a vertical line at \( x = 2 \) with domain \( x = 2 \) and range \( y \in (-\infty, \infty) \).

Step by step solution

01

Rewrite the equation in slope-intercept form

First, let's rewrite the equation \[ -3x + 6 = 0 \] in slope-intercept form which is \[ y = mx + b\] form.
02

Solve for x

Rearrange the equation to solve for x: down_arrow_down_arrow\[-3x + 6 = 0\] \Rightarrow & \Rightarrow \ \[ -3x = -6 \ \ x = 2 \]
03

Determine the type of line

The equation \(x = 2\) represents a vertical line. Vertical lines run parallel to the y-axis and have an undefined slope.
04

Graph the line

To graph the line, draw a vertical line passing through \ \( x = 2\) on the coordinate plane.
05

Determine the domain

The domain of this vertical line is simply \(x = 2\) since it does not extend in the x-direction.
06

Determine the range

The range of a vertical line is all possible values of \(y\), since it extends infinitely in the y-direction. Thus, the range is \( y \in (-\infty, \infty) \).

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Key Concepts

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

slope-intercept form
The slope-intercept form is a common way to express linear equations. It is represented as \( y = mx + b \). In this format, \( m \) is the slope of the line and \( b \) is the y-intercept, where the line crosses the y-axis. To transform any linear equation into this form, you'll want to isolate \( y \) on one side of the equation.
If an equation is given in a different format, like \( -3x + 6 = 0 \), you need to manipulate it to make it match \( y = mx + b \). In this example, though, the equation simplifies to \( x = 2 \), which is different from typical slope-intercept equations.

This is because the simplified equation \( x = 2 \) represents a vertical line, which doesn't have a slope or y-intercept.
domain and range
Understanding the domain and range of a function is essential in graphing.
The domain of a function is all the input values (x-values) that the function can accept. For most lines, the domain is all real numbers because you can plug any x-value into the equation and get a y-value.
However, for vertical lines like \( x = 2 \), the domain is limited to a single x-value. Thus, the domain of \( x = 2 \) is \( x = 2 \).

The range of a function is all the possible output values (y-values). Vertical lines have no limitations in the y-direction. They extend infinitely up and down. Therefore, the range of a vertical line is all real numbers, which we denote as \( y \in (-\backslash \backslash{infty}, \backslash \backslash{infty}) \).
vertical lines
Vertical lines are unique in their properties when compared to other lines. They have an equation in the form \( x = a \), where \( a \) is a constant. This means the line runs parallel to the y-axis at \( a \).
One of the key characteristics of vertical lines is that they have an undefined slope. This is because the change in x-values is zero, and division by zero is not defined.
When you graph a vertical line, you draw a line that intersects the x-axis at the given x-value and runs vertically.

In summary, vertical lines have:
  • An undefined slope
  • A domain restricted to a single x-value
  • A range of all possible y-values

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Most popular questions from this chapter

Let \(f(x)=x^{2}+3\) and \(g(x)=-2 x+6 .\) Find each of the following. $$(f g)(-3)$$

The tables give some selected ordered pairs for functions \(f\) and \(g\). $$\begin{array}{|c|c|c|c|}\hline x & 3 & 4 & 6 \\\\\hline f(x) & 1 & 3 & 9 \\\\\hline\end{array}$$ $$\begin{array}{|c|c|c|c|c|}\hline x & 2 & 7 & 1 & 9 \\\\\hline g(x) & 3 & 6 & 9 & 12 \\\\\hline\end{array}$$ Find each of the following. $$(f \circ g)(7)$$

Consumption Expenditures In Keynesian macroeconomic theory, total consumption expenditure on goods and services, \(C\), is assumed to be a linear function of national personal income, \(I\). The table gives the values of \(C\) and \(I\) for 2004 and 2009 in the United States (in billions of dollars). $$\begin{array}{|c|c|c|}\hline \text { Year } & 2004 & 2009 \\\\\hline \text { Total consumption (C) } & \$ 8285 & \$ 10,089 \\\\\hline \text { National income (I) } & \$ 9937 & \$ 12,026 \\\\\hline\end{array}$$ (a) Find the formula for \(C\) as a function of \(I\) (b) The slope of the linear function is called the marginal propensity to consume. What is the marginal propensity to consume for the United States from \(2004-2009 ?\)

Find functions \(f\) and \(g\) such that \((f \circ g)(x)=h(x) .\) (There are many possible ways to do this.) See Example 9 $$h(x)=(2 x-3)^{3}$$

Find the function \(g(x)=a x+b\) whose graph can be obtained by translating the graph of \(f(x)=2 x+5\) up 2 units and to the left 3 units.
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