An Introduction to Linear Algebra by L. Mirsky

By L. Mirsky

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A function whose graph is symmetric with respect to the origin is an odd function. A graph that is symmetric with respect to the x-axis is not the graph of a function ͑except for the graph of y ϭ 0͒. 30. y y y (x , y ) (−x, y) Graph each function with a graphing utility. Determine whether the function is even, odd, or neither. 30 (x, −y) Symmetric to origin Odd function Symmetric to x-axis Not a function p͑x͒ ϭ x9 ϩ 3x5 Ϫ x3 ϩ x What do you notice about the equations of functions that are (a) odd and (b) even?

B) Use the result of part (a) to complete the table. Ϫ10Њ C F 0Њ 10Њ 177Њ 68Њ 90Њ 95. Business A contractor purchases a bulldozer for $36,500. 50 per hour. (a) Write a linear equation giving the total cost C of operating the bulldozer for t hours. ) (b) Assuming that customers are charged $65 per hour of bulldozer use, write an equation for the revenue R derived from t hours of use. (c) Use the profit formula ͑P ϭ R Ϫ C͒ to write an equation for the profit gained from t hours of use. (d) Use the result of part (c) to find the break-even point (the number of hours the bulldozer must be used to gain a profit of 0 dollars).

To sketch the graph of a piecewise-defined function, you need to sketch the graph of each equation on the appropriate portion of the domain. Example 8 Sketching a Piecewise-Defined Function Sketch the graph of f ͑x͒ ϭ ΆϪx2x ϩϩ 4,3, You use a graphing utility to graph f ͑x͒ ϭ Ά x2 ϩ 1, 4 Ϫ x, x Յ 0 x > 0 by letting y1 ϭ x2 ϩ 1 and y2 ϭ 4 Ϫ x, as shown in the figure. You conclude that this is the graph of f. What’s wrong? x Յ 1 x > 1 7 by hand. Solution −6 This piecewise-defined function is composed of two linear functions.

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