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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) \(f \circ g\) and (b) \(g \circ f .\) Find the domain of each function and each composite function. $$f(x)=\sqrt[3]{x-5}, \quad g(x)=x^{3}+1$$

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