Math 19: Calculus Winter 2008 Instructor: Jennifer KlokeLecture OutlineWednesday, January 9Theory: Interval NotationOften we have to write down intervals, or combinations of intervals, on the real line. For example,we might write something like ”the domain is all x ≥ 0.” However, there is a more convenientnotation that is usually used. This notation is best explained by example.Note: Parentheses correspond to < and > and square brackets correspond to ≤ and ≥.For multiple intervals, use theSsymbol to combine intervals.Examples: Interval Notation1. Write −1 ≤ x < 2 or 2 < x < 3 or x ≥ 4 in interval notation.2. What is the domain of y = ln(x − 3)?3. What is the domain of y =1x(x−2)?Theory: Piecewise FunctionsA piecewise function is a function given by different formulas on different intervals. For example,f(x) =x2: x < 0x : 0 ≤ x < πsin x : x ≥ πis a piecewise function that is equal to x2when x is negative, equal to x when x is in the interval[0, π), and equal to sin x when x ≥ π.Examples: Piecewise Functions1. Graph the following piecewise function:f(x) =ln x : 0 < x < e1 : e ≤ x < 56 − x : x ≥ 52. Be able to write down a piecewise function corresponding to a graph.1Theory: Function CompositionAn important way of putting two functions together is function composition:(f ◦ g)(x) = f(g(x)).To evaluate f ◦ g at x, first evaluate g at x, and then plug this value into f
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