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Chapter 1: Problem 49

Determine whether the statement is true or false. Justify your answer. A piecewise-defined function will always have at least one \(x\) -intercept or at least one \(y\) -intercept.

Short Answer

Expert verified
The statement is false. Not all piecewise-defined functions have to have an x-intercept or a y-intercept.

Step by step solution

01

Understanding intercepts

An x-intercept is the point where the curve of the graph crosses or touches the x-axis. The intercept will have coordinates in the form of \( (x, 0) \) since at that point, \( y = 0 \). Similarly, a y-intercept is the point where the curve of the graph crosses or touches the y-axis. Its coordinates will be in the form of \( (0, y) \), because at that point, \( x = 0 \).
02

Break down of Piecewise function

A piecewise function is a function which is defined by multiple sub-functions, each pertaining to a certain interval of the independent variable. While each of these sub-functions can individually have their own intercepts, it's not necessary for all piecewise functions to have an intercept. It is possible to create a piecewise function that doesn't have either x-intercept or y-intercept.
03

Concluding the statement

Based on this reasoning, it can be concluded that the statement given 'A piecewise-defined function will always have at least one x-intercept or at least one y-intercept' is false. It鈥檚 possible for a piecewise function to exist that does not intercept either axis.

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Most popular questions from this chapter

Find a mathematical model that represents the statement. (Determine the constant of proportionality.) \(F\) is jointly proportional to \(r\) and the third power of \(s\) \((F=4158 \text { when } r=11 \text { and } s=3 .)\)

Sketch the graph of the function. $$f(x)=\left\\{\begin{array}{ll}1-(x-1)^{2}, & x \leq 2 \\\\\sqrt{x-2}, & x>2\end{array}\right.$$

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Finding a Mathematical Model In Exercises \(41-50\), find a mathematical model for the verbal statement. Newton's Law of Universal Gravitation: The gravitational attraction \(F\) between two objects of masses \(m_{1}\) and \(m_{2}\) is jointly proportional to the masses and inversely proportional to the square of the distance \(r\) between the objects.

The percents \(p\) of prescriptions filled with generic drugs in the United States from 2004 through 2010 (see figure) can be approximated by the model \(p(t)=\left\\{\begin{array}{ll}4.57 t+27.3, & 4 \leq t \leq 7 \\ 3.35 t+37.6, & 8 \leq t \leq 10\end{array}\right.\) where \(t\) represents the year, with \(t=4\) corresponding to \(2004 .\) Use this model to find the percent of prescriptions filled with generic drugs in each year from 2004 through \(2010 .\) (Source: National Association of Chain Drug Stores) (GRAPH CAN'T COPY)
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