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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

A company produces a product for which the variable cost is 12.30 dollars per unit and the fixed costs are 98,000 dollars. The product sells for 17.98 dollars. Let \(x\) be the number of units produced and sold. (a) The total cost for a business is the sum of the variable cost and the fixed costs. Write the total cost \(C\) as a function of the number of units produced. (b) Write the revenue \(R\) as a function of the number of units sold. (c) Write the profit \(P\) as a function of the number of units sold. (Note: \(P=R-C\) ).

Fill in the blanks. An ordered pair of real numbers can be represented in a plane called the rectangular coordinate system or the ____ plane.

Fill in the blanks. The domain of \(f\) is the ____of \(f^{-1},\) and the ____of \(f^{-1}\) is the range of \(f\).

The cost per unit in the production of an MP3 player is 60 dollars. The manufacturer charges 90 dollars per unit for orders of 100 or less. To encourage large orders, the manufacturer reduces the charge by 0.15 dollars per MP3 player for each unit ordered in excess of 100 (for example, there would be a charge of 87 dollars per MP3 player for an order size of 120 ). (a) The table shows the profits \(P\) (in dollars) for various numbers of units ordered, \(x .\) Use the table to estimate the maximum profit. $$\begin{array}{|l|c|c|c|c|c|}\hline \text { Units, } x & 130 & 140 & 150 & 160 & 170 \\\\\hline \text { Profit, } P & 3315 & 3360 & 3375 & 3360 & 3315 \\\\\hline\end{array}$$ (b) Plot the points \((x, P)\) from the table in part (a). Does the relation defined by the ordered pairs represent \(P\) as a function of \(x ?\) (c) Given that \(P\) is a function of \(x,\) write the function and determine its domain. (Note: \(P=R-C\) where \(R\) is revenue and \(C\) is cost.)

During a nine-hour snowstorm, it snows at a rate of 1 inch per hour for the first 2 hours, at a rate of 2 inches per hour for the next 6 hours, and at a rate of 0.5 inch per hour for the final hour. Write and graph a piecewise- defined function that gives the depth of the snow during the snowstorm. How many inches of snow accumulated from the storm?
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