(1/30) MA160/MA161 : Semester 1 Calculus Week 3: Functions (domain, range, piecewise, odd and even) http://www.maths.nuigalway.ie/~niall/MA161-1 22nd and 23rd of September, 2014 Tutorials (2/30) MA160 Tutorials started last week: Thursday at 13.00 in IT125G. ..................................................................... MA161 Tutorials start from today (19th September). For those registered for MA161 should attend one of: 1 Monday at 14:00 in IT203. 2 Tuesday at 09:00 in AM105 3 Tuesday at 11:00 in IT206 4 Tuesday at 18:00 in AC201 5 Wednesday at 18:00 in AM104 6 Friday at 11:00 in IT203 Assignments (3/30) MA161 Set “Problem Set 1” Deadline: 5pm, Friday 03 Oct 2014 Corresponds to: Calculus and Algebra from Weeks 1–3. Number of questions: 15 Max attempts: 10 Access at: Blackboard. See also: www.maths.nuigalway.ie/∼niall/MA161-1 Credit: 5% of MA161. ..................................................................... MA160 Set “Problem Sheet 1” Deadline: 5pm, Friday 26 September, 2014 Corresponds to: Calculus and Algebra from Weeks 1–3. Number of questions: 10 Max attempts: 10 Access at: http://brnsrv.nuigalway.ie:8128 Credit: 4.5% of MA160 (approximately). MA161 Problem Set 1 (4/30) All MA161 homework assignments are run using an on-line system called WeBWorK. It assigns a unique problem set for each student. Everyone has the same number and type of exercises, but the numbers/coefficients are different for every one. You access the homework by logging onto Blackboard and selecting the module 1415-MA161, and then choosing the assignment: Outline 1 Recall... Domain, co-domain and range 2 Piecewise functions The absolute value function Other examples 3 Even and Odd functions 4 1. Linear functions 5 2. Polynomials 6 Exercises (5/30) Recall... Domain, co-domain and range (6/30) Recall that a function is a rule that maps values from one set to another. In MA16x, we are mainly concerned with functions from R to R. Given the formula for a function, we’ll frequently have to figure out What is the domain of f ? That is the set of all numbers x ∈ R, such that f (x) makes sense; What is the range of f ? That is the set of all numbers y ∈ R, such that y = f (x) for some x. When figuring out the domain, we need to take into account: for what values of R for which f defined. In particular, we must avoid dividing by zero. function f (x) = 1/x is not defined at x = 0. for what values of R does f map to a real number? For example, √ x = x 1/2 is a real number only when x ≥ 0; Recall... Domain, co-domain and range (7/30) Example What (subset of R) are the largest possible domain and range for the √ function f (x) = x + 2? Recall... Domain, co-domain and range (8/30) Example (MA160 Problem Sheet 1, Question 8) What is the domain of the following real-valued function? f (x) = 17 . x 3 + 2x 2 − 8x Recall... Domain, co-domain and range (9/30) Example: Suppose that f is a function f : X → Y given by f (x) = 1 + x 3 . 1 If the domain X is the set [0, ∞), what this the range? 2 What domain gives the range [0, ∞)? Piecewise functions The absolute value function (10/30) So far, we have met inequality expressions in solving problems like Let f (x) = x 3 − 4x. For what values of x is f (x) ≥ 0? However, we will often have functions where inequality expressions are used in the definition of the function. These are piecewise-defined functions. The most important example of this is the absolute value function: Absolute value The absolute value of a real number x, denoted as |x| is: ( −x x < 0 |x| = x x ≥ 0. Piecewise functions The absolute value function (11/30) Examples: Sketch the graph of the following functions: A. f (x) = |x| B. f (x) = |x − 1| C. f (x) = |x| − 1 Piecewise functions Examples: The Heavyside function... Other examples (12/30) Piecewise functions Other examples (13/30) Example: Sketch the graph of the piecewise defined function ( 1 + x x < −1 f (x) = 1 2 x ≥ −1. 2x Solution 1 f (x) 0 −1 −2 −3 −3 −2 −1 x 0 1 Piecewise functions Other examples (14/30) A piecewise defined function that you might be familiar with is the one that maps scores in Leaving Cert exams to CAO points. Lets consider the mapping for higher-level papers: f : [0, 100] → [0, 100] Grade A1 A2 B1 B2 B3 C1 C2 C3 D1 D2 D3 E F NG Score 90 ≤ x ≤ 100 85 ≤ x < 90 80 ≤ x < 85 75 ≤ x < 80 70 ≤ x < 75 65 ≤ x < 70 60 ≤ x < 65 55 ≤ x < 60 50 ≤ x < 55 45 ≤ x < 50 40 ≤ x < 45 25 ≤ x < 40 10 ≤ x < 25 0 ≤ x < 10 Points 100 90 85 80 75 70 65 60 55 50 45 0 0 0 Piecewise functions Other examples (15/30) Example: Sketch both the functions f (x) = x 3 3 1 0 −1 −2 −3 −2 −1 0 x 1 2 g (x) = |x 3 |. 8 7 6 5 4 3 2 1 0 −1 −2−1 0 1 2 x g(x) = |x3 | f (x) = x3 2 and Piecewise functions Other examples (16/30) Example (From MA161 Problem Set 1, Question 2) Solve the inequality: Even and Odd functions Definition A function f is called even if f (−x) = f (x). A function f is called odd if f (−x) = −f (x). It is possible for a function to be neither odd nor even. Example Show that the function f (x) = x 2 is even. (17/30) Even and Odd functions Example Show that the function f (x) = x3 − x is odd. x2 + 1 (18/30) Even and Odd functions Example 1 2 Is the function f (x) = |x| odd or even? Is the function f (x) = |3 − x| odd or even? (19/30) Even and Odd functions (20/30) The notion of “even” and “odd” functions is closely related to symmetry . An even function is symmetric about the y -axis, while an odd function is symmetric about the point (0, 0). Examples: 3 1 g(x) = |x3 | f (x) = x3 2 0 −1 −2 −3 −2 −1 8 7 6 5 4 3 2 1 0 −1 −2−1 0 1 2 x 0 x 1 2 Even and Odd functions (21/30) Example: (MA160/MA161 Paper 1, 2012/13, Q4(c)) For each of the following functions, determine if it is odd, even or neither: 1 (i) f (x) = + x 7 + x 9 ; x (ii) g (x) = −5x 2 − 3x 4 − 2. A catalogue of functions (22/30) [The following material follows Section 1.2 of Stewart’s Calculus quite closely .] We’ll now recap some of the most common functions that you’ll meet during the course of the year. They include 1. Linear functions; 2. Polynomials; 3. Power functions; 4. Rational functions; 5. Algebraic functions; 6. Trigonometric functions; 7. Exponential functions; 8. Logarithms. 0. Trivial functions But first we’ll look at the most boring function known to science: f : R → R give by f (x) = 0 for all x. Only marginally more exciting are the constant functions, e.g., f : R → R give by f (x) = 5 for all x. (23/30) 1. Linear functions (24/30) A linear function is one whose graph is a straight line. Its equation is given by l(x) = mx + b, where m is the slope, and b is the y -intercept. Examples: Graphing linear functions is quite easy: Examples: 2. Polynomials (25/30) Slightly more exciting than the linear functions are the polynomials A function P is called a polynomial if its equation has the form: P(x) = an x n + an−1 x n−1 + · · · + a2 x 2 + a1 x + a0 , where n is a nonnegative integer, and a0 , a1 , . . . , an are real numbers called the coefficients. If an 6= 0 then we say that “P has degree n”. Examples: 2. Polynomials (26/30) Other than constants and linear functions, the most important examples of polynomial functions are the quadratics and cubics: Examples: 2. Polynomials (27/30) Example: Here is the graph of a cubic polynomial. What is its equation? 3 2 y 1 0 −1 −2 −2 −1 0 x 1 2 3 Exercises (28/30) Q1. Find the domain and largest possible range for the following functions, and sketch them: (i) f (x) = x|x| (ii) f (x) = 1 |x| + 1 (iii) f (x) = 1 |x + 1| ( (iv) f (x) = x +1 if x < 0 1−x if x ≥ 0. x + 9 (v) f (x) = −2x −6 if x < −3 if |x| ≤ 3 if x > 3. Exercises (29/30) Q2. Solve the following inequality: |2x + 5| + 20 ≥ 25. Q3. For each of the following functions, determine if it is even, odd, or neither. x x2 + 1 x2 f (x) = 4 x +1 f (x) = x|x| t 3 + 3t f (t) = 4 t − 3t 2 + 4 f (x) = 2 + x 2 + x 4 (i) f (x) = (ii) (iii) (iv) (v) Q4. Are the trigonometric functions sin, cos and tan even, odd, or neither? Exercises (30/30) Q5. Here is the graph of a quadratic polynomial. What is its equation? Q6. The following graph is of a cubic polynomial whose equation is y = K (x − a)(x − b)(x − c). 4 3 Find a, b, c and K . 2 y 1 2 0 1 −1 0 −2 y −1 −3 −4 −3 −2 −1 0 x 1 2 3 −2 −3 −3 −2 −1 0 x 1 2 3

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