M1410105Part3

M1410105Part3 - (Section 1.5: Analyzing Graphs of...

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(Section 1.5: Analyzing Graphs of Functions) 1.58 PART H: GRAPHING PIECEWISE-DEFINED FUNCTIONS Example: See p.69 for the graph of f x ( ) = x   or x . It is a piecewise constant function (in particular, a step function) with a piecewise horizontal graph. See Section 1.4, Part H: Notes 1.32 . Example Graph our function f from Section 1.4, Part G: Notes 1.31 , where f is defined by: f x ( ) = x 2 , 2 x < 1 x + 1, 1 x 2 Its graph is below:
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(Section 1.5: Analyzing Graphs of Functions) 1.59 The top rule for f x ( ) corresponds to the parabolic piece on the left. The left endpoint at x = 2 is a filled-in circle, because it is included on the graph by way of the “weak” symbol. The right endpoint at x = 1 is a hollow circle, because it is excluded from the graph by way of the “strict” < symbol. It helps to know that, at x = 1 , x 2 would have been 1 in value, so that we place this circle at the point 1,1 ( ) . In a way, for this purpose we are “borrowing” the top rule for the case x = 1 , even though it technically does not apply. The bottom rule corresponds to the line segment on the right. Both endpoints are filled-in circles, because those points are included on the graph by way of the symbols. In Calculus: When we study limits and continuity in Calculus ( Chapter 2 in the Math 150 textbook at Mesa ), we often study graphs with breaks like the one at x = 1 in the previous Example; that break is called a jump discontinuity. Technical Note: We could not allow 2 x 1 to be the domain for the x 2 rule, given that 1 x 2 is the domain for the x + 1 rule, because f 1 ( ) would then not be well defined; it would be ambiguous. The VLT would have failed. In our Example, it is unambiguous that f 1 ( ) = 2 , based on the bottom rule. However, remember that we did “borrow” the top rule for the purposes of locating the hollow circle at x = 1 .
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(Section 1.5: Analyzing Graphs of Functions) 1.60 PART I: THE HORIZONTAL LINE TEST (HLT) (Also see Section 1.9 .) In Part B, Notes 1.40 , we discussed the Vertical Line Test (VLT). An equation in x and y describes x as a function of y , and we can then say x = f y ( ) Its graph passes the HLT in the standard xy -plane, meaning that there is no horizontal line that intersects the graph more than once (i.e., there is no input
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M1410105Part3 - (Section 1.5: Analyzing Graphs of...

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