Week 1 Course Notes

# Week 1 Course Notes - MATH 137 WEEK 1 NOTES PROF DOUG PARK...

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MATH 137 WEEK 1 NOTES PROF. DOUG PARK 1. Functions Notations. R = { real numbers } Z = { integers } = { 0 , ± 1 , ± 2 ,... } N = { +’ve integers } = { 1 , 2 , 3 ,... } Interval Notation: [ a,b ] = { x R | a x b } [ a,b ) = { x R | a x < b } ( a,b ] = { x R | a < x b } ( a,b ) = { x R | a < x < b } The last item is not to be confused with the point ( a,b ) R 2 . Notation works with . [ a, ) = { x R | x a } ( -∞ ,b ) = { x R | x < b } ( -∞ , ) = R Deﬁnitions. A function f : A B is a rule (formula) that assigns to each element x in a set A exactly one element f ( x ) in a set B . A is called the domain of f . The set f ( A ) = { f ( x ) | x A } is called the range of f . (Here, means “is an element of”.) Unless otherwise speciﬁed, the domain of f will be the biggest possible set of x values on which the formula for f ( x ) makes sense. 2. Power functions Graphs. Here are some graphs you should already know. y - b = m ( x - a ) Line through the point ( a,b ) with slope m Date : September 18, 2009. 1

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2 DOUG PARK m ( a , b ) 1 Ex . y = x 2 n ( n = 1 , 2 , 3 ,... ) even function, f ( - x ) = f ( x ), i.e. ( - x ) 2 n = x 2 n . Graph of even function is symmetric about y -axis ( x = 0). Generally, the larger the n , the graph goes to faster and also goes to 0 faster! 0 x y y=x (1,1) (-1,1) 2 y=x 4 Ex . y = x 2 n +1 ( n = 1 , 2 , 3 ,... ) odd function, f ( - x ) = - f ( x ), i.e. ( - x ) 2 n +1 = - x 2 n +1 . Graph of odd function is symmetric about the origin. increasing function, a < b implies f ( a ) < f ( b ), i.e. a 2 n +1 < b 2 n +1 Increasing function always has an inverse : f ( x ) = x 2 n +1 , f - 1 ( x ) = x 1 2 n +1 0 x y y=x (1,1) (-1,-1) 3 y=x 5
MATH 137 3 Ex . y = x 1 2 n +1 = 2 n +1 x ( n = 1 , 2 , 3 ,... ) As the inverse function of x 2 n +1 , the graph is the reﬂection of y = x 2 n +1 along the line y = x . 0 x y y=x (1,1) (-1,-1) 1/3 y=x 1/5 Ex . y = x 1 2 n = 2 n x ( n = 1 , 2 , 3 ,... ) Domain = { x | x 0 } = Range. x 2 n is not 1–1, but x 2 n restricted to x 0 is 1–1. As the inverse of this restriction, the graph is the reﬂection of the right half of y = x 2 n along y = x . 0 x y y=x (1,1) 1/2 y=x 1/4 y= -x Ex . y = 1 x 2 n = x - 2 n ( n = 1 , 2 , 3 ,... ) Domain = { x | x 6 = 0 } and Range = { x | x > 0 } . even function

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Week 1 Course Notes - MATH 137 WEEK 1 NOTES PROF DOUG PARK...

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