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Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 6. A major complaint of professors teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2-semester) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course | 1.3 Representations of Functions 31 Specific Example General Statement Symbolic Representation P is a square if and only if P is a rectangle with sides of equal length A if and only if B A B P is a square if P is a rectangle with sides of equal length A if B equivalently equivalently If P is a rectangle with sides of equal length then P is a square if B then A B A P is a square only if P is a rectangle with sides of equal length A only if B equivalently equivalently If P is a square then P is a rectangle with sides of equal length if A then B A B The graphs of some functions are given in the figure below. f V a f V A Figure 1.16 height volume 2 c the calibration function b A r - PV C V for the evaporating flask Graphs of mappings Q R and S from Exercise 1.1 are given in Figure 1.17. You should now be able to check visually that Q and R are functions while S is not a function. Q 16 - 3 Q 3 3 6 15 14 - 1m C 12 - 111 -c--1---1--1--1---1-- _ . 1 3 6 a 3 R 3 3 TC 1 3 1 1 -2 -4 -6 b Figure 1.17 c 5 6 4 2 32 CHAPTER 1 Functions Are Lurking Everywhere EXERCISE 1.3 Two of the four graphs given in Figure 18 are the graphs of functions.15 Which two are they Can you come up with a rule for determining whether or not a given graph is the graph of a function Answer We can tell that the relationships represented in Figures 1.16 a - c 1.17 a and b and 1.18 b and d are in fact functions. The test for a function is that every input must have only one output assigned to it graphically this means if we draw a vertical line through any point on the horizontal axis this line cannot cross the graph in more than one place. The vertical line will not cross the graph at all if it is drawn through a point on the horizontal axis that is not a valid input i.e. not in the domain of the function. This test for deciding if a graph represents a function is called the vertical line test. y d A x Let s return to the problem of calibrating bottles. When we calibrate a bottle we put in a known volume
