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Chapter 0: Problem 19

Determine whether the statement is true or false. Explain your answer. A curve that crosses the \(x\) -axis at two different points cannot be the graph of a function.

Short Answer

Expert verified
False. A curve can cross the x-axis at two points and still be a function if it passes the vertical line test.

Step by step solution

01

Understanding the Problem

The problem asks us to determine whether a curve that crosses the \(x\)-axis at two different points can be the graph of a function. For a curve to represent a function, it must pass the vertical line test: any vertical line should intersect the graph at most once.
02

Apply the Vertical Line Test

To determine if the graph is a function, draw several vertical lines across different parts of the graph. If any vertical line intersects the graph more than once, the graph is not a function. In this scenario, having a curve cross the \(x\)-axis at two different points does not necessarily imply failure of the vertical line test.
03

Consider Example of a Polynomial Function

Polynomial functions like quadratic functions can cross the \(x\)-axis at two different points, for example, \(y = (x-a)(x-b)\), where \(a\) and \(b\) are distinct real numbers. This function crosses the \(x\)-axis at \(x = a\) and \(x = b\), but it remains a function because any vertical line intersects it at most once.
04

Conclusion from Vertical Line Test

Since the curve crosses the \(x\)-axis at two different points and still can be a function if it satisfies the vertical line test (such as in the quadratic example), the original statement is false. A curve crossing the \(x\)-axis twice can still be a graph of a function.

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Key Concepts

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

Vertical Line Test
The Vertical Line Test is a simple tool used to determine if a curve on a graph is a function. To apply this test, imagine drawing vertical lines across the graph at various positions.
If any vertical line crosses the graph more than once, the graph is not a function. This is because a function can only output one value (y-value) for each input (x-value).
  • If every vertical line crosses the graph only once, the curve is a function.
  • If any vertical line crosses the graph twice or more, the curve is not a function.
Understanding this allows us to see that even if a curve crosses the x-axis multiple times, it may still pass the vertical line test and represent a valid function.
Polynomial Functions
Polynomial functions are some of the most commonly studied functions in mathematics. They can be expressed in the form of an equation, where the exponents of the variables are whole numbers.
For example, a polynomial function could look like this:
  • Quadratic Function: \( f(x) = ax^2 + bx + c \)
  • Cubic Function: \( f(x) = ax^3 + bx^2 + cx + d \)
Polynomial functions are known for their smooth and continuous graphs. It's important to note that the function may cross the x-axis at one or more points, known as real roots.
In the quadratic function example, if it's in the form \( y = (x-a)(x-b) \), the graph could cross the x-axis at \( x = a \) and \( x = b \) while still being a valid function.
X-axis Intersections
X-axis intersections in graphing represent the points where the graph crosses the x-axis. These intersections are significant because they indicate the real roots or solutions of the function.
When a curve crosses the x-axis, its y-value equals zero at that particular x-value. A polynomial function, for example, might cross the x-axis multiple times without violating the definition of a function.
  • The graph of \( y = (x-a)(x-b) \) is one where intersections occur at different points on the x-axis.
  • Despite these crossings, the function can remain valid as long as it passes the Vertical Line Test.
This means that x-axis intersections are permissible for functions, as long as each x-value corresponds to only one y-value on the curve.

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Most popular questions from this chapter

A variable \(y\) is said to be imversely proportional to the square of a variable \(x\) if \(y\) is related to \(x\) by an equation of the form \(y=k / x^{2}\), where \(k\) is a nonzero constant, called the constant of proportionality. This terminology is used in these exercises. According to Coulomb's law, the force \(F\) of attraction between positive and negative point charges is inversely proportional to the square of the distance \(x\) between them. (a) Assuming that the force of attraction between two point charges is \(0.0005\) newton when the distance between them is \(0.3\) meter, find the constant of proportionality (with proper units). (b) Find the force of attraction between the point charges when they are 3 meters apart. (c) Make a graph of force versus distance for the two charges. \(\quad(c a=r .)\) (d) What happens to the force as the particles get closer and closer together? What happens as they get farther and farther apart?

Express \(f\) as a composition of two functions; that is, find \(g\) and \(h\) such that \(f=g \circ h .\) [Note: Each exercise has more than one solution.] (a) \(f(x)=x^{2}+1\) (b) \(f(x)=\frac{1}{x-3}\)

Find formulas for \(f \circ g\) and \(g \circ f\), and state the domains of the compositions. $$ f(x)=\frac{1+x}{1-x}, g(x)=\frac{x}{1-x} $$

The perceived loudness \(\beta\) of a sound in decibels \((\mathrm{dB})\) is related to its intensity \(I\) in watts per square meter \(\left(\mathrm{W} / \mathrm{m}^{2}\right)\) by the equation $$ \beta=10 \log \left(I / I_{0}\right) $$ where \(I_{0}=10^{-12} \mathrm{~W} / \mathrm{m}^{2}\). Damage to the average ear occurs at \(90 \mathrm{~dB}\) or greater. Find the decibel level of each of the following sounds and state whether it will cause ear damage. $$ \begin{array}{ll} \hline \text { SOUND } & {I} \\ \hline \text { (a) Jet aircraft (from 50 ft) } & 1.0 \times 10^{2} \mathrm{~W} / \mathrm{m}^{2} \\ \text { (b) Amplified rock music } & 1.0 \mathrm{~W} / \mathrm{m}^{2} \\ \text { (c) Garbage disposal } & 1.0 \times 10^{-4} \mathrm{~W} / \mathrm{m}^{2} \\ \text { (d) TV (mid volume from 10 ft) } & 3.2 \times 10^{-5} \mathrm{~W} / \mathrm{m}^{2} \\ \hline \end{array} $$

Suppose that the function \(f\) has domain all real numbers. Determine whether each function can be classified as even or odd. Explain. (a) \(g(x)=\frac{f(x)+f(-x)}{2}\) (b) \(h(x)=\frac{f(x)-f(-x)}{2}\)
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