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Chapter 3: Problem 74

Use a graphing utility. Consider the equation $$y=\left\\{\begin{array}{ll}1 & \text { if } x \text { is rational } \\\0 & \text { if } x \text { is irrational }\end{array}\right.$$ Is this a function? What is its domain? What is its range? What is its \(y\) -intercept, if any? What are its \(x\) -intercepts, if any? Is it even, odd, or neither? How would you describe its graph?

Short Answer

Expert verified
Yes, it's a function. Domain: \(\mathbb{R}\), Range: {0, 1}, y-intercept: (0, 1), Infinitely many x-intercepts. Even function.

Step by step solution

01

Verify if it's a function

To determine if the given equation defines a function, recall that a function assigns exactly one output for every input. The given equation assigns a value of 1 or 0 depending on whether x is rational or irrational, respectively. For any input x, it gives one unique output (either 1 or 0), so it is a function.
02

Determine the domain

The domain of a function is the set of all possible input values. In this case, the inputs are any real numbers (since any real number can be either rational or irrational). Thus, the domain is all real numbers, \(\mathbb{R}\).
03

Determine the range

The range of a function is the set of all possible output values. For this function, the output is either 1 (when x is rational) or 0 (when x is irrational). Therefore, the range is {0, 1}.
04

Identify the y-intercept

The y-intercept is the point where the graph crosses the y-axis. This occurs when x=0. Since 0 is a rational number, the output is y=1. Therefore, the y-intercept is (0, 1).
05

Identify the x-intercepts

The x-intercepts are points where the graph crosses the x-axis, which happens when y=0. For this to occur, x must be irrational. Therefore, any irrational number serves as an x-intercept since the function evaluates to 0 for such inputs. Hence, there are infinitely many x-intercepts.
06

Determine if the function is even, odd, or neither

A function is even if \(f(-x)=f(x)\) for every x in the domain. It's odd if \(f(-x)=-f(x)\). Here, \(f(x)\) is either 1 or 0 based on the rationality of x. \(f(-x)\) will give 1 or 0 based on whether -x is rational or irrational. Since rational numbers are symmetric about the origin (same for irrationals), \(f(-x)\) equals \(f(x)\). Therefore, the function is even.
07

Describe the graph

The graph consists of a set of points at y=1 for all x that are rational and y=0 for all x that are irrational. This creates two horizontal lines, one at y=1 and one at y=0, though these lines are not continuous since they are made up of discrete points scattered along the x-axis based on the rationality or irrationality of each x value.

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Key Concepts

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

domain and range
In mathematics, the domain and range are two key concepts related to the functionality of specific equations. The domain of a function refers to all the possible input values that the function can accept. For the given equation, any real number is a valid input. This includes both rational and irrational numbers. Therefore, the domain of the function is the set of all real numbers, denoted as \(\textbf{R}\).

The range of a function is the set of all possible output values. For our function, the output can only be either 1 or 0, depending on whether the input is rational or irrational. Thus, the range is \(\text\{0, 1\}\).
This means the function can only output two values, making it unique and straightforward in terms of what it can achieve.
y-intercept
The y-intercept of a function is the point where the graph crosses the y-axis. This occurs when x = 0. In our function, we must evaluate the function at x = 0 to find the y-intercept. Since 0 is a rational number, our function assigns the value 1 to y. Therefore, the y-intercept is the point \(0, 1\).

This point is particularly significant because it tells us where the function touches the y-axis, which can often serve as a starting point or reference when graphing the function.
even functions
In mathematics, a function is considered even if it satisfies the condition that \(f(-x) = f(x)\) for all x in its domain. Our function is defined based on the rationality of x. Both rational and irrational numbers are symmetric around the origin
  • meaning if x is rational, -x is also rational.
  • Similarly, if x is irrational, -x is also irrational.
This symmetry means that \(f(-x)\) will have the same value as \(f(x)\), either 1 or 0. Therefore, the function is even.
This property is critical as even functions have a symmetrical graph about the y-axis. In our case, since the values switch between 1 and 0 depending on rationality, the graph still reflects this symmetry.
x-intercept
The x-intercepts of a function are the points where the graph crosses the x-axis, which is when y = 0. For our function, y = 0 when x is irrational. Thus, every irrational number is an x-intercept.
  • This means there are infinitely many x-intercepts scattered along the x-axis wherever irrational numbers exist.

Irrational numbers like \(\text\{ 蟺, e, \sqrt\{2\} \}\) are common examples of x-intercepts in our function. These points on the graph show where the output dips to 0, especially important for visualizing how the function behaves with different input types.
real numbers
Real numbers are a fundamental concept in mathematics, encompassing every number that can appear on a number line. This includes:
  • Rational numbers such as 5, -2.5, and 0
  • Irrational numbers such as \pi (蟺), \(e\), and \(\sqrt{2}\)

In our function, the domain is composed entirely of real numbers. Depending on whether a real number is rational or irrational, the output changes between 1 and 0. This makes real numbers a versatile and all-inclusive category for the inputs in this function.

Understanding real numbers helps in grasping other concepts such as the domain and intercepts, providing a complete picture of function behavior.

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Most popular questions from this chapter

\(G(x)=-x^{4}+32 x^{2}+144\) (a) Determine whether \(G\) is even, odd, or neither. (b) There is a local maximum value of 400 at \(x=4\). Find a second local maximum value. (c) Suppose the area of the region enclosed by the graph of \(G\) and the \(x\) -axis between \(x=0\) and \(x=6\) is 1612.8 square units. Using the result from (a), determine the area of the region enclosed by the graph of \(G\) and the \(x\) -axis between \(x=-6\) and \(x=0\).

For the function \(f(x)=x^{2},\) compute the average rate of change: \(\begin{array}{ll}\text { (a) From } 1 \text { to } 2 & \text { (b) From } 1 \text { to } 1.5\end{array}\) (c) From 1 to 1.1 (d) From 1 to 1.01 (e) From 1 to 1.001 (f) Use a graphing utility to graph each of the secant lines along with \(f\) (g) What do you think is happening to the secant lines? (h) What is happening to the slopes of the secant lines? Is there some number that they are getting closer to? What is that number?

Graph each function using the techniques of shifting, compressing, stretching, and/or reflecting. Start with the graph of the basic function (for example, \(y=x^{2}\) ) and show all the steps. Be sure to show at least three key points. Find the domain and the range of each function. $$ g(x)=3|x+1|-3 $$

\(g(x)=x^{2}-2\) (a) Find the average rate of change from -2 to 1 . (b) Find an equation of the secant line containing \((-2, g(-2))\) and \((1, g(1))\).

Two cars are approaching an intersection. One is 2 miles south of the intersection and is moving at a constant speed of 30 miles per hour. At the same time, the other car is 3 miles east of the intersection and is moving at a constant speed of 40 miles per hour. (a) Build a model that expresses the distance \(d\) between the cars as a function of time \(t\). [Hint: At \(t=0,\) the cars are 2 miles south and 3 miles east of the intersection, respectively.] (b) Use a graphing utility to graph \(d=d(t) .\) For what value of \(t\) is \(d\) smallest?
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