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M1410106

# M1410106 - (Section 1.6 A Library of Parent Functions 1.70...

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(Section 1.6: A Library of Parent Functions) 1.70 SECTION 1.6: A LIBRARY OF PARENT FUNCTIONS PART A: CONSTANT FUNCTIONS If f x ( ) = c , where c is some real number, then f is a constant function. Any real input yields the same output, c . If f x ( ) = 3 , for example, we have: Although it is not acceptable for a function to have the same (legal) input yield multiple outputs, it is acceptable to have multiple inputs yielding the same output. This is not true of one-to-one functions, however; we will study these in Section 1.9 . A constant function f is even, since: x R , f x ( ) = c , and f x ( ) = c , and it has a flat horizontal graph. If f x ( ) = 3 , for example, our graph for y = 3 is in purple below: Warning: f x ( ) = 3 , for example, represents an even function, although 3 is an odd integer. Don’t confuse the issue of even and odd functions with even and odd integers.

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(Section 1.6: A Library of Parent Functions) 1.71 PART B: THE IDENTITY FUNCTION The odd function f x ( ) = x is called the identity function, because its output is identical to its input. Some input-output machines: 3 f 3 10 f → − 10 Its graph is below: Technical Note: This function comes up in the discussion of inverse functions ( Section 1.9 ).
(Section 1.6: A Library of Parent Functions) 1.72 PART C: “BOWLS” In Section 1.5: Notes 1.51-1.52 and p.67 , we discussed the graph of f x ( ) y = x 2 . Because f is an even function, its graph is symmetric about the y -axis: The graph above for the x 2 function is called a parabola, which is a type of conic section we will see again in Section 2.1 and Section 10.2 . The graphs for x 2 , x 4 , x 6 , etc. never fall below the x -axis, because the corresponding functions are never negative in value. The graphs for x 4 , x 6 , x 8 , etc. are also symmetric about the y -axis (because they also correspond to even functions) and have similar “bowl” shapes, but they are not parabolas.

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(Section 1.6: A Library of Parent Functions) 1.73 PART D: THE “TWILIGHT ZONE” (This was actually the title of a classic TV series in which the strange and unexpected happened.) In fact, the strange and unexpected often happens when comparing graphs of basic functions on the intervals 0,1 ( ) and perhaps 1,0 ( ) . For example, we will compare the graph for the
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M1410106 - (Section 1.6 A Library of Parent Functions 1.70...

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