09.09 - [Collect students' notes on 1.2. Allow one day's...

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[Collect students’ notes on 1.2. Allow one day’s grace: “You can hand them in on Monday.”] [Remind them that the homework on section 1.1 is due on MONDAY.] The graph of y = f ( x ) = 1/sqrt(1– x 2 ) looks like 2 1 1 2 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 Is the function f ( x ) even? ..?. . Yes. The graph of y = g ( x ) = 1/sqrt( x 2 –1) looks like
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2 1 1 2 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 Is the function g ( x ) even? ..?. . Yes. What does the graph of y = f ( x ) + g ( x ) = 1/sqrt(1– x 2 ) + 1/sqrt( x 2 –1) look like? ..?. . It’s “pointless”! 2 1 1 2 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 Note that we sometimes want to work with functions that aren’t defined everywhere, like f ( x ) and g ( x ). But then we’re led to include functions like f ( x )+ g ( x ) that aren’t
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defined ANYWHERE. Mathematicians prefer to be inclusive in their notion of what counts as a function, so the “empty” function is considered a function, even though it’s pretty boring. Is the function f ( x ) + g ( x ) even? . .. ..?. . Call the sum h ( x ). What is the truth-status of the proposition “For all x in the domain of h , – x is in the domain of h and satisfies h (– x ) = h ( x )”? (Recall that this proposition is precisely the meaning of the assertion “
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09.09 - [Collect students' notes on 1.2. Allow one day's...

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