Functions: Piecewise, Even and Odd.

Transcription

1 Functions: Piecewise, Even and Odd. MA161/MA1161: Semester 1 Calculus. Prof. Götz Pfeiffer School of Mathematics, Statistics and Applied Mathematics NUI Galway September 21-22, 2015

2 Tutorials, Online Homework. Tutorials start from today (September 21). You should attend one of the following: 1. Monday at 14:00 in ENG-2002, 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 1-3, 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 real-valued 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 y-ais. 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 y-intercept. Eample f() = 2 1. Slope: 2. y-intercept: 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