1_Functions.pdf

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Department
Mathematics
Course
MATA30H3
Professor
Sophie Chrysostomou
Semester
Winter

Description
FUNCTIONS 1.1 FUNCTIONS DEFINITION: A function, f, with domain D, is a rule that associates, to each element x ∈ D a single real number, y or f(x). • The domain of f is often written as dom(f). Also, x ∈ dom(f) is often expressed as f is defined at x. • The range of a function f is the set of all of the function values. range(f) = {y ∈R | y = f(x) for some x ∈ D}. • When the function is given but its domain is not mentioned, we assume the domain of this function to be the largest subsRtfor which this function can be defined. √ EXAMPLE: f(x) = 2 − x 8 ▯2013 by Sophie Chrysostomou 1.1 FUNCTIONS The Vertical Line Test : This test is used to determine if a relation is a function. A relation is a function if and only if there are no vertical lines that intersect the graph at more than one point. Functions can be represented by an arrow diagram, by words, by a table of values, by an algebraic rule or by a graph. EXAMPLE of a function represented by a graph: 2.5 -2.5 0 2.5 Thus, f has domain all the values in the interval and range all the values in the interval . These can be read directly from the graph. 9 ▯2013 by Sophie Chrysostomou FUNCTIONS DEFINITION: If f is a function f and f(−x) = f(x) for all x ∈ dom(f), then f is called an even function. If f(−x) = −f(x), for all x ∈ dom(f) then f is called an odd function EXAMPLE: Determine if each of the following functions is odd, even or neither. 2 4 4 2 5 x − x (a) h(x) = 5x + x (b) g(x) = x − 5 (c) f(x) = x + x 3 HOMEWORK: Show that: (i) the product of two even functions is an even one. (ii) the product of two odd functions is an even one. (iii) the product of an odd and an even function is an odd one. 10 ▯2013 by Sophie Chrysostomou 1.1 FUNCTIONS DEFINITION: Let f(x) be a function defined on an interval I. We say f is (i) increasing on I if for all x ,x ∈ I, 1 2 x1< x ⇒2(x ) < f1x ) 2 (ii) decreasing on I if for all x ,x ∈ I, 1 2 x1< x ⇒2(x ) > f1x ) 2 (iii) monotonic if on its domain it is everywhere increasing or everywhere decreasing. (The interval I may be of any of the forms [a,b],(a,b),(a,b],[a,b),(−∞,b),(a,∞), (−∞,b],[a,∞),(−∞,∞)). 11 ▯2013 by Sophie Chrysostomou FUNCTIONS 1.2 A LIBRARY OF FUNCTIONS 1. The Constant Function: A function of the form f(x) = C, where C is a constant number, is called a constant function. p 2. A Power Function: A function is of the form f(x) = kx where k and p are real constants with k ▯= 0. Here are some examples of power functions: (a) Even Power Functions: p = 2n where n = 1,2,3,..... Graphs of some even power functions: 2n 5 f(x) = x , n = 1,2,3 x^6 4 x^4 Domain is 3 Range is 2 x^2 In each of the intervals 1 (−∞,−1), (−1,0), (0,1), (1,∞). observe which function is the maximum -1 0 1 and which is the minimum. 12 ▯2013 by Sophie Chrysostomou 1.2 A LIBRARY OF FUNCTIONS (b) Odd Power Functions: p = 2n − 1 where n = 1,2,3,..... Graphs of some odd power functions: 2n−1 3 f(x) = x n = 1,2,3 x^5 x^3 2 Domain is x 1 Range is -1 0 1 In each of the intervals -1 (−∞,−1), (−1,0), (0,1), (1,∞). observe which function is the maximum -2 and which is the minimum. -3 13 ▯2013 by Sophie Chrysostomou FUNCTIONS 3. The n-th Root Functions: These are functions of the form √n 1 f(x) = ( x) = x (where n is a positive integer) The rule here depends on the parity of n. (The parity of a number indicates whether the number is even or odd.) • If n is even, the rule is that y = f(x) is the unique positive number such that y = x. Its domain and range are the same: [0,∞). • If n is odd, the rule is that y = f(x) is the unique number such that y = x. Its domain and range are the same: (−∞,∞). Example: 1 f(x) = x has domain and range 1 f(x) = x 3 has domain and range Graphs of Even Root Functions Graphs of Odd Root Functions 1 1.6 0.8 -1.6 -0.8 0 0.8 1.6 2.4 3.2 4 -0.8 0 0.8 1.6 2.4 3.2 -1 14 ▯2013 by Sophie Chrysostomou 1.2 A LIBRARY OF FUNCTIONS 4. Polynomials: A polynomial P of degree n, where n is a non-negative integer, is a function given by a formula of the form P(x) = a xn+ a n−1 xn−1 + ▯▯▯ + a0 where a n...,a ∈0 R are constants, and a ▯n 0. The domain of any polynomial is R . The even and odd power functions are examples of polynomials. The function f(x) = 0 is also considered to be a polynomial, but it has no degree. It is called the zero polynomial. EXAMPLE: Determine which of the following are polynomials: f(x) = 3x − 2x 2 g(x) = 5x + πx − 11 π h(x) = 5x + 3x − 11 15 ▯2013 by Sophie Chrysostomou FUNCTIONS The following theorems help us to factor polynomials: Rational Root theorem: If a polynomial f(x) = a x + a n n n−1 xn−1 + ▯▯▯ + a 0 a n= p with integral coefficients, i.e. a ∈ i Z , has a nonzero rational root in its lowest terms then p is a divisor of a and q is a q 0 divisor of a . n Factor theorem: Let P(x) denote a polynomial of degree n ≥ 1 and c a real number. Then x − c is a factor of P(x) if and only if P(c) = 0, that is, if and only if c is a root of P(x). EXAMPLE: Factor f(x) = x 3− x − x − 2 HOMEWORK: 1) Use the factor theorem to show that (a) x − 2x + 4x − 5x − 6 is divisible by x − 2. 5 4 3 2 2 (b) x + 4x + 2x − 2x + x − 6 is divisible by x + 2x − 3. 3 2 (c) x + (2 − a)x + (5 − 2a)x − 5a is divisible by x − a. 4 3 2 2) Use the rational root theorem to find if p(x) = x +8x +19x +22x+10 has any rational roots. If there are such roots factor p(x) as far as possible. 16 ▯2013 by Sophie Chrysostomou 1.2 A LIBRARY OF FUNCTIONS 5. Rational Functions: A rational function is just the ratio of two polynomials. For example if P and Q are polynomials (and Q is not the zero polynomial) then P f = Q P(x) is the rational function whose value at x is given by the rule f(x) = Q(x) Note that the largest domain of f is the set of all x ∈ R such that Q(x) ▯= 0. EXAMPLE: Find the largest possible domain of the rational function f(x) = 3x − 2 x − 1 17 ▯2013 by Sophie Chrysostomou FUNCTIONS Remainder Theorem: Let f(x) = P(x) be a rational function (with Q(x) P(x) R(x) P and Q polynomials). Then f(x) = = S(x) + where S Q(x) Q(x) and R are polynomials with degR < degQ. (This is basically long or synthetic division.) x − 4x − x + 5x + 1 EXAMPLE: f(x) = 5 4 x − 2x − x + 2 P(x) HOMEWORK: For the following functions f(x) = express f(x) D(x) R(x) in the form f(x) = Q(x) + , where degR < degD. D(x) x2 x + 1 x3 i) f(x) = x − 1 ii) f(x) = x − 2 iii) f(x) =x − 2 3 3 3 iv) f(x) = x − b v) f(x) = x − 3 vi) vi) f(x) = x − 1 x + a x + x + 1 x + x + 1 18 ▯2013 by Sophie Chrysostomou 1.2 A LIBRARY OF FUNCTIONS 6. Piecewise Defined Functions: A piecewise-defined function f(x) of a real variable x is a function whose definition is given differently on disjoint subsets of its domain. (Disjoint sets have no intersection.) EXAMPLE: Let    2x if x < −5 f(x) = x + 1 if −5 ≤ x < 5  −2x if x > 5 (a) Find f(−10),f(−1),f(3),f(7) . (b) Find the domain of f. 19 ▯2013 by Sophie Chrysostomou FUNCTIONS Examples of piecewise defined functions: (a) The Absolute Value Function: This is the function | ▯ | given by the formula:  |x| = x if x ≥ 0  −x if x < 0 Note that the values of the absolute value function are always greater than or equal to zero. The graph is This function has domain and range √ Note |x| = x2. 0 Three important properties of the absolute value function are: (i) |xy| = |x||y| (ii) |x + y| ≤ |x| + |y| (iii) ||x| − |y|| ≤ |x − y| 20 ▯2013 by Sophie Chrysostomou 1.2 A LIBRARY OF FUNCTIONS (b) The Floor and Ceiling Functions: These two functions are de- fined by ⌊x⌋ = the largest integer ≤ x ⌈x⌉ = the smallest integer ≥ x HOMEWORK: Give the graph of the ceiling function. 7. Exponential Functions: For a fixed positive number a ▯= 1, the function f(x) = a is called an exponential function with base a . For this function, domf = R and range f = (0,∞). If a > 1 we have exponential growth: If a < 1 we have exponential decay: 0 0 Every exponential growth function eventually dominates every power function. So for any a > 1 and any n, for large enough values of x, a > x . n 21 ▯2013 by Sophie Chrysostomou FUNCTIONS LAWS OF EXPONENTS If a, b are positive numbers and x, y are any real numbers, then: (1.) a = 1 (2.) ax+y = a ay x −x 1 a x−y (3.) a = x (4.) y= a a a (5.) (ab) = a b x (6.) (a ) = a xy  −1 2 0−1  x y z  EXAMPLE: Simplify 3 −4 2 x y z HOMEWORK: Assuming that all variables represent positive real numbers only, simplify each of the following expressions as much as possible so that all exponents are positive.  −1 2 0−1  2 −2  2 −1 −1  x y z   a b c  x + y (a.) x y z 2 (b.) a b c 0 (c.) (xy)−1 . 22 ▯2013 by Sophie Chrysostomou 1.2 A LIBRARY OF FUNCTIONS 8. Trigonometric Functions: The definition of trigonometric functions depends on the following ge- ometric diagram, on which θ is considered to be positive if measured counterclockwise from the x-axis. (x,y) sinθ = cscθ = 1 ! cosθ = secθ = 0 tanθ = cotθ = Using right angle trigonometry, we can use the triangle below, which is similar to the triangle in the unit circle above, to get: sinθ = cscθ = cosθ = secθ = tanθ = cotθ = hyp = hypotenuse, adj = adjacent, opp = opposite 23 ▯2013 by Sophie Chrysostomou FUNCTIONS The graphs of sinθ, cosθ, tanθ are: 24 ▯2013 by Sophie Chrysostomou 1.2 A LIBRARY OF FUNCTIONS
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