Functions: Piecewise, Even and Odd.


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1 Functions: Piecewise, Even and Odd. MA161/MA1161: Semester 1 Calculus. Prof. Götz Pfeiffer School of Mathematics, Statistics and Applied Mathematics NUI Galway September 2122, 2015
2 Tutorials, Online Homework. Tutorials start from today (September 21). You should attend one of the following: 1. Monday at 14:00 in ENG2002, 2. Tuesday at 09:00 in AM107, 3. Tuesday at 11:00 in IT206, 4. Tuesday at 18:00 in AC201, 5. Wednesday at 18:00 in AM104, 6. Thursday at 13:00 in IT125G (MA1161 only), 7. Friday at 10:00 in AC Friday at 11:00 in AC204.
3 MA161 Problem Set 1. Deadline: 5pm, Friday, October 02, Covers: Weeks 13, Calculus and Algebra. Length: 15 Questions. Attempts: 10. Access: via blackboard. See also: Credit: 5%.
4 Recall... Recall that a function is a rule that maps values from one set to another. In this course, we are mainly concerned with functions f: D R, where D R. Given the formula for a function f, we frequently have to figure out: What is the domain of f? (The domain of f is the set of all numbers R such that f() makes sense.) What is the range of f? (The range of f is the set of all numbers y R such that y = f() for some in the domain of f.) When figuring out the domain of f, we need to take into account: For which values is f() defined? In particular, we must avoid dividing by zero. The function f() = is not defined at = 3. For which values does f map to a real number? For eample, 3 is a real number only if 3 0.
5 Eamples. Eample What (subsets of R) are the largest possible domain and range for the function f() = + 2? Eample (MA160 (2014) Problem Sheet 1, Question 8) What is the domain of the following realvalued function? f() = Eample Suppose that f: D R is a function given by f() = If the domain D is the interval [0, ), what is the range? 2. What domain gives the range [0, )?
6 The Absolute Value Function. Inequalities can be used inside the definition of a function. This gives a piecewise defined function. The most important eample of a piecewise defined function is the absolute value function. Definition (Absolute Value) The absolute value of a real number, denoted as, is {, if 0, =, else. Eamples 17 = ( 17) = = < 1 = (2, 4).
7 Piecewise Defined Functions. Eample (The Heaviside Function) H(t) = { 0, if t < 0, 1, if t 0. Eample Sketch the graph of the piecewise defined function { 1 +, if < 1, f() = 1 2 2, if 1. Eample (MA161 Problem Set 1, Question 2) Solve the inequality > 10.
8 Cubes. Eample (f() = 3 vs. g() = 3.) y y
9 Definition A function f: D R is called even if f( ) = f() for all in its domain D. A function f: D R is called odd if f( ) = f() for all in its domain D. Most functions are neither even nor odd. Eample Show that the function f() = 2 is even. Show that the function f() = 3 2 is odd. + 1 Eample 1. Is the function f() = even or odd? 2. Is the function f() = 3 odd or even?
10 Symmetries. The notion of even and odd is related to symmetries. Any even function is symmetric about the yais. An odd function is symmetric about the origin (0, 0). Eamples (f() = 3 vs. g() = 3.) y y
11 Even or Odd? Eample (MA160/MA161 Paper 1, 2012/13) For each of the following functions, determine if it is even, odd or neither: 1. f() = ; 2. g() =
12 A Catalog of Functions. [Section 1.2 of the Book] We ll now spend some time reviewing the most common essential functions. These include: 1. Linear Functions; 2. Polynomials; 3. Power Functions; 4. Rational Functions; 5. Algebraic Functions; 6. Trigonometric Functions; 7. Eponential Functions; 8. Logarithms.
13 0. Constant Functions. But first, we look at the simplest possible functions. Let c R. The constant function f() = c assigns the same value c to all R. Eample (The graph of f() = 3) y y = f() The constant function f() = 0 is the only function f: R R which is even and odd at the same time...
14 1. Linear Functions. A linear function is one whose graph is a straight line. It can be represented by a formula of the form f() = m + b, where m is the slope, and b is the yintercept. Eample f() = 2 1. Slope: 2. yintercept: 1. f(1) = 1, f( 1) = 3,... y y = f() Linear functions are easy to graph!
15 Eercises. 1. Find the largest possible domain and range for the following functions and sketch them: (i) f() =, (iii) f() = 1 + 1, + 9, if < 3, (v) f() = 2, if 3, 6, if > Solve the following inequality: (ii) f() = 1 + 1, { + 1, if < 0, (iv) f() = 1, if 0, 3. For each of the following functions, determine if it is even, odd, or neither. (i) f() = 2 + 1, (iii) f() =, (v) f() = t3 + 3t t 4 3t (ii) f() = , (iv) f() =
16 Eercises. 4. Are the trigonometric functions sin, cos, and tan even, odd, or neither? 5. Here is the graph of a quadratic polynomial. What is its equation? 6. The following graph is that of a cubic polynomial with equation y 1 y = f() f() = K( a)( b)( c). Find a, b, c, and K. 1 y y = f() 1 1
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