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Chapter 1: Problem 74

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 that 'A piecewise-defined function will always have at least one \(x\) -intercept or at least one \(y\) -intercept' is True.

Step by step solution

01

Understanding Piecewise-defined Functions

Often a function can be defined by different expressions or formulas in different ranges of the \(x\)-values. Such functions are called piecewise-defined functions. For example, the absolute value function \(f(x) = |x|\) is defined by \(f(x) = x\) for \(x 鈮� 0\) and \(f(x) = -x\) for \(x < 0\).
02

Understanding \(x\)-intercepts and \(y\)-intercepts

In the graphic representation of a function, an \(x\)-intercept is the point at which the function intersects the \(x\)-axis. This means, for some value of \(x\), \(f(x) = 0\). Conversely, a \(y\)-intercept is the point at which the function intersects the \(y\)-axis. This will always happen when \(x = 0\).
03

Evaluating the Statement

Consider a constant function, which is also a piecewise function, such as \(f(x) = 2\). This function has no \(x\)-intercept because \(f(x)\) can never equal 0. However, it does have one \(y\) -intercept at (0, 2). Similarly, consider a function such as \(f(x) = x + 3\) for \(x \geq 3\) and \(f(x) = x - 3\) for \(x < 3\). This function does not have a \(y\)-intercept as it is never defined at \(x = 0\) but it does have an \(x\)-intercept at \(x = 3\). Therefore, the statement that a piecewise function will always have at least one \(x\) -intercept or one \(y\) -intercept is true.

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Most popular questions from this chapter

Write a sentence using the variation terminology of this section to describe the formula. Volume of a sphere: \(V=\frac{4}{3} \pi r^{3}\)

Find a mathematical model for the verbal statement. \(V\) varies directly as the cube of \(e\).

Use the given value of \(k\) to complete the table for the direct variation model $$y=k x^{2}$$ Plot the points on a rectangular coordinate system. $$\begin{array}{|l|l|l|l|l|l|} \hline x & 2 & 4 & 6 & 8 & 10 \\ \hline y=k x^{2} & & & & & \\ \hline \end{array}$$ $$ k=1 $$

The total numbers of people (in thousands) in the U.S. civilian labor force from 1992 through 2007 are given by the following ordered pairs. $$\begin{array}{ll} (1992,128,105) & (2000,142,583) \\ (1993,129,200) & (2001,143,734) \\ (1994,131,056) & (2002,144,863) \\ (1995,132,304) & (2003,146,510) \\ (1996,133,943) & (2004,147,401) \\ (1997,136,297) & (2005,149,320) \\ (1998,137,673) & (2006,151,428) \\ (1999,139,368) & (2007,153,124) \end{array}$$ A linear model that approximates the data is \(y=1695.9 t+124,320,\) where \(y\) represents the number of employees (in thousands) and \(t=2\) represents 1992 . Plot the actual data and the model on the same set of coordinate axes. How closely does the model represent the data? (Source: U.S. Bureau of Labor Statistics)

Two techniques for fitting models to data are called direct _____ and least squares _____ .
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