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# 3. Functions II.pdf

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# 3. Functions II.pdf

Properties of Functions
Odd and Even Functions
Periodic Functions
Monotonic Functions
Bounded Functions
Maxima and Minima of Functions
Inverse Function
Sequence and Series

Properties of Functions
Odd and Even Functions
Periodic Functions
Monotonic Functions
Bounded Functions
Maxima and Minima of Functions
Inverse Function
Sequence and Series

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### 3. Functions II.pdf

1. 1. FUNCTIONS II Dr. Gabriel Obed Fosu Department of Mathematics Kwame Nkrumah University of Science and Technology Google Scholar: https://scholar.google.com/citations?user=ZJfCMyQAAAAJ&hl=en&oi=ao ResearchGate ID: https://www.researchgate.net/profile/Gabriel_Fosu2 Dr. Gabby (KNUST-Maths) Functions 1 / 41
2. 2. Lecture Outline 1 Properties of Functions Odd and Even Functions Periodic Functions Monotonic Functions Bounded Functions Maxima and Minima of Functions 2 Inverse Function 3 Sequence and Series Dr. Gabby (KNUST-Maths) Functions 2 / 41
3. 3. Properties of Functions Odd and Even Functions Even Function Let f be a function and Df its domain. We assume that if x ∈ Df then −x ∈ Df . Definition (Even Function) f is an even function if f (−x) = f (x). Example The functions f (x) = x2 , g(x) = −x4 + 2x2 − 1, h(x) = cos(x) + x2 , i(x) = x sinx are even functions since: 1 f (−x) = (−x)2 = x2 = f (x), 2 g(−x) = −(−x)4 +2(−x)2 −1 = g(x), 3 h(−x) = cos(−x)+(−x)2 = h(x), and 4 i(−x) = (−x)sin(−x) = −x(−sinx) = x sinx = i(x). Dr. Gabby (KNUST-Maths) Functions 4 / 41
4. 4. Properties of Functions Odd and Even Functions Odd Function Definition (Odd Function) f is an odd function if f (−x) = −f (x). Example The functions f (x) = x, f (x) = −x3 + 2x, f (x) = sin(x), and f (x) = csc(x), f (x) = tan(x) are odd functions. Because 1 f (−x) = −x = −f (x) 2 f (−x) = −(−x)3 +2(−x) = x3 −2(x) = −f (x) 3 f (−x) = sin(−x) = −sin(x) = −f (x) 4 f (−x) = 1 sin(−x) = − 1 sin(x) = −f (x) Dr. Gabby (KNUST-Maths) Functions 5 / 41
5. 5. Properties of Functions Odd and Even Functions Remarks The graph of an even function is symmetric about the y-axis. The graph of an odd function is symmetric about the origin. Dr. Gabby (KNUST-Maths) Functions 6 / 41
6. 6. Properties of Functions Periodic Functions Periodic Functions Definition Let f be a function, and Df its domain, then f is a periodic function if there exists a positive real number t such that f (x + t) = f (x) for all x ∈ Df . The minimum of such t’s which is often denoted as T , is called the period of f . Example The trigonometric functions are periodic functions. 1 sin(x +2kπ) = sin(x +2π) = sin(x) for k ∈ Z, however, T = 2π. 2 cos(x +2kπ) = cos(x +2π) = cos(x),T = 2π. 3 tan(x +(2k +1)π) = tan(x +π) = tan(x),T = π. Dr. Gabby (KNUST-Maths) Functions 7 / 41
7. 7. Properties of Functions Monotonic Functions Monotonic Functions Let I be an open interval. x1 and x2 are two elements of I such that x1 < x2. Definition ☛ f is an increasing function on I if f (x1)<f (x2). ☛ f is a decreasing function on I if f (x1)>f (x2). Example • The functions ex , tan(x) and ax + b, where a > 0, are increasing on their respective domains. • The functions e−x , cot(x) and ax + b, where a < 0, are decreasing on their respective domains. Dr. Gabby (KNUST-Maths) Functions 8 / 41
8. 8. Properties of Functions Monotonic Functions Monotonic Decreasing Function −5 −4 −3 −2 −1 1 2 3 4 5 −3 −2 −1 1 2 3 0 −3x +1 ↘ e−x ↘ Dr. Gabby (KNUST-Maths) Functions 9 / 41
9. 9. Properties of Functions Monotonic Functions Monotonic Increasing Function −5 −4 −3 −2 −1 1 2 3 4 5 −3 −2 −1 1 2 3 0 2x −1 ↗ ex ↗ Dr. Gabby (KNUST-Maths) Functions 10 / 41
10. 10. Properties of Functions Monotonic Functions Monotonic Increasing and Decreasing −5 −4 −3 −2 −1 1 2 3 4 5 −3 −2 −1 1 2 3 0 sinx ↗↘ Dr. Gabby (KNUST-Maths) Functions 11 / 41
11. 11. Properties of Functions Monotonic Functions Monotonic Functions Example Show that the function f (x) = p x −2 is an increasing function on its domain. 1 Df = [2,+∞). 2 For x1,x2 ∈ Df and x1 < x2, 3 2 < x1 < x2 =⇒ 0 < x1 −2 < x2 −2 4 =⇒ 0 < p x1 −2 < p x2 −2 5 =⇒ f (x1) < f (x2). 6 Thus, f is an increasing function on its domain. Dr. Gabby (KNUST-Maths) Functions 12 / 41
12. 12. Properties of Functions Monotonic Functions Monotonic Functions Example Show that f (x) = (2− x)2 +1 decreases on (−∞,2] and increases on [2,+∞). 1 Df = R. 2 For x1,x2 ∈ (−∞,2], 3 x1 < x2 ≤ 2 =⇒ −x1 > −x2 > −2 4 =⇒ 2− x1 > 2− x2 > 0 5 =⇒ (2− x1)2 > (2− x2)2 > 0 6 =⇒ (2− x1)2 +1 > (2− x2)2 +1 > 1 7 =⇒ f (x1) > f (x2). 8 f is decreasing on (−∞,2]. 1 For x1,x2 ∈ [2,+∞), 2 2 ≤ x1 < x2 =⇒ −2 > −x1 > −x2 3 =⇒ 0 > 2− x1 > 2− x2 4 =⇒ 0 < (2− x1)2 < (2− x2)2 5 =⇒ 1 < (2− x1)2 +1 < (2− x2)2 +1 6 =⇒ f (x1) < f (x2). 7 f is an increasing function on [2,+∞). Dr. Gabby (KNUST-Maths) Functions 13 / 41
13. 13. Properties of Functions Bounded Functions Bounded Functions Definition A function is said to be bounded above if there is ū ∈ R such that f (x) ≤ ū for all x in the domain of f . Example The function f (x) = x2 + 1 defined on 0 ≤ x ≤ 1 is bounded above by 2 since f (x) ≤ 2 for 0 ≤ x ≤ 1. Example The function f (x) = 1/x defined on x ∈ N is bounded above by 1 Example The function f (x) = sinx is bounded above by 1 for x ∈ R. Dr. Gabby (KNUST-Maths) Functions 14 / 41
14. 14. Properties of Functions Bounded Functions Bounded Functions Definition A function, f , is said to be bounded below if there is ℓ ∈ R such that f (x) ≥ ℓ for all x in the domain of f . Example The function f (x) = x−1 defined in [0,1] is bounded below by −1 since −1 ≤ f (x) for x ∈ [0,1]. Example The function g(x) = | p x +1| is bounded below by 0 on the interval [0,4] since 0 ≤ g(x) for x ∈ [0,4]. Dr. Gabby (KNUST-Maths) Functions 15 / 41
15. 15. Properties of Functions Maxima and Minima of Functions Maxima and Minima of Functions Local(or relative) and Global(or absolute) Minimum 1 The function f is said to have a local minimum value at the point x0 if f (x0) ≤ f (x) for all x in a neighbourhood of x0. 2 f is said to have a global minimum value at the point x0 if f (x0) ≤ f (x) for all x in the domain of f . 3 In this case f is bounded below. Local(or relative) and Global(or absolute) Maximum 1 If f (x) ≤ f (x0) for all x in a neighbourhood of x0, then f has a local maximum value at the point x0. 2 The maximum is global if f (x) ≤ f (x0) for all x in the domain of f . 3 In this case f is bounded above. Dr. Gabby (KNUST-Maths) Functions 16 / 41
16. 16. Properties of Functions Maxima and Minima of Functions Maxima and Minima of Functions −5 −4 −2 2 4 −3 −1 1 2 3 0 global min f (x) = 1+(x +1)2 , Dr. Gabby (KNUST-Maths) Functions 17 / 41
17. 17. Properties of Functions Maxima and Minima of Functions Maxima and minima −5 −4 −2 2 4 −3 −1 1 2 3 0 local max local min g(x) = 1−2x −3x2 +2x3 Dr. Gabby (KNUST-Maths) Functions 18 / 41
18. 18. Properties of Functions Maxima and Minima of Functions Maxima and minima −5 −4 −2 2 4 −3 −1 1 2 3 0 global min local max local min h(x) = (x −1)(−x +3)2 (x), Dr. Gabby (KNUST-Maths) Functions 19 / 41
19. 19. Inverse Function Inverse Functions 1 An inverse function is a function that undoes the action of the another function. 2 A function g is the inverse of a function f if whenever y = f (x) then x = g(y) 3 In other words, applying f and then g is the same thing as doing nothing. We can write this in terms of the composition of f and g as g(f (x)) = x 4 A function f has an inverse function only if for every y in its range there is only one value of x in its domain for which f (x) = y 5 This inverse function is unique and is frequently denoted by f −1 and called f inverse. Dr. Gabby (KNUST-Maths) Functions 21 / 41
20. 20. Inverse Function Inverse Functions Given the function f (x) we want to find the inverse function, f −1 (x) 1. First, replace f (x) with y. 2. Replace every x with a y and replace every y with an x. 3. Solve the equation from Step 2 for y. This is the step where mistakes are most often made so be careful with this step. 4. Replace y with f −1 (x). In other words, we’ve managed to find the inverse at this point! 5. Verify your work by checking that (f ◦ f −1 )(x) = x and (f −1 ◦ f )(x) = x are both true. Example Given f (x) = 3x −2 find f −1 (x) Dr. Gabby (KNUST-Maths) Functions 22 / 41
21. 21. Inverse Function 1 Given the function f f (x) = 3x −2 2 First, replace f (x) with y. y = 3x −2 3 Replace every x with a y and replace every y with an x. x = 3y −2 4 Solve the equation from Step 2 for y. y = 1 3 (x +2) 5 Replace y with f −1 (x). f −1 (x) = x 3 + 2 3 Dr. Gabby (KNUST-Maths) Functions 23 / 41
22. 22. Inverse Function We can verify the results, we check that (f ◦ f −1 )(x) = x (f ◦ f −1 )(x) = f [f −1 (x)] = f hx 3 + 2 3 i = 3 hx 3 + 2 3 i −2 = x +2−2 = x Dr. Gabby (KNUST-Maths) Functions 24 / 41
23. 23. Inverse Function Graph of a Function and Its Inverse There is an interesting relationship between the graph of a function and the graph of its inverse. Here is the graph of the function and inverse from the first two examples. In both cases the graph of the inverse is a reflection of the actual function about the line y = x . This will always be the case with the graphs of a function and its inverse. Dr. Gabby (KNUST-Maths) Functions 25 / 41
24. 24. Sequence and Series Sequence and Series Definition A sequence is an ordered set of numbers that most often follows some rule (or pattern) to determine the next term in the order. Example x,x2 ,x3 ,x4 , ... is a sequence of numbers, where each successive term is multiplied by x. Definition A series is a summation of the terms of a sequence. The greek letter sigma Σ is used to represent the summation of terms of a sequence of numbers. Dr. Gabby (KNUST-Maths) Functions 27 / 41
25. 25. Sequence and Series Sequence and Series Series are typically written in the following form: n X i=1 ai = a1 + a2 + a3 ···+ an where the index of summation, i takes consecutive integer values from the lower limit, 1 to the upper limit, n. The term ai is known as the general term. Example 5 X i=1 i = 1+2+3+4+5 = 15 (1) 6 X k=3 2k = 23 +24 +25 +26 = 8+16+32+64 = 120 (2) 4 X k=1 kk = 11 +22 +33 +44 = 1+4+27+256 = 288 (3) Dr. Gabby (KNUST-Maths) Functions 28 / 41
26. 26. Sequence and Series Finite Series Definition (Finite and Infinite Series) A finite series is a summation of a finite number of terms. An infinite series has an infinite number of terms and an upper limit of infinity. Properties of Finite Series The following are the properties for addition/subtraction and scalar multiplication of series. For some sequence ai and bi and a scalar k then: n X i=1 kai ±bi = k n X i=1 ai ± n X i=1 bi Dr. Gabby (KNUST-Maths) Functions 29 / 41
27. 27. Sequence and Series Theorems of Finite Series 1 The following theorems give formulas to calculate series with common general terms. These formulas, along with the properties listed above, make it possible to solve any series with a polynomial general term, as long as each individual term has a degree of 3 or less. n X i=1 1 = n (4) n X i=1 c = nc (5) n X i=1 i = n(n +1) 2 (6) n X i=1 i2 = n(n +1)(2n +1) 6 (7) n X i=1 i3 = µ n(n +1) 2 ¶2 (8) (9) Dr. Gabby (KNUST-Maths) Functions 30 / 41
28. 28. Sequence and Series Types of Sequences There are two main types of sequences. 1 An arithmetic sequence is one in which successive terms differ by the same amount. For example, {3, 6, 9, 12, ···}. Note each term is obtained by adding 3 to the previous term. This is called the common difference denoted as d = 6−3 2 A geometric sequence is one in which the quotient of any two successive terms is a constant. For example, {3, 9, 27, 81, ···}. Note each term is obtained by multiplying the previous term by 3. This is called the common ratio denoted by r = 9 3 . Similarly, there are also arithmetic series and geometric series, which are simply summations of arithmetic and geometric sequences, respectively. Dr. Gabby (KNUST-Maths) Functions 31 / 41
29. 29. Sequence and Series Theorems for Arithmetic and Geometric Series Arithmetic Series Suppose we have the following arithmetic series, {a +(a +d)+(a +2d)+···+(a +(n −1)d} Then, n−1 X k=0 a +kd = n 2 (2a +(n −1)d) (10) Geometric Series Suppose we have the following geometric series, {a + ar + ar2 +...+ ar(n−1) } Then, n−1 X k=0 ark = a µ rn −1 r −1 ¶ (11) Dr. Gabby (KNUST-Maths) Functions 32 / 41
30. 30. Sequence and Series Example Solve the series 20 X i=1 2i2 +7i 20 X i=1 2i2 +7i = 20 X i=1 2i2 + 20 X i=1 7i = 2 µ 20(20+1)(2(20)+1) 6 ¶ +7 µ 20(20+1) 2 ¶ = 2 µ 17220 6 ¶ +7(210) = 5740+1470 = 7210 Dr. Gabby (KNUST-Maths) Functions 33 / 41
31. 31. Sequence and Series Example Find the sum of the geometric series: 2+8+32+128+...+8192 1 We know that 8 2 = 4 and that 32 8 = 4, so r = 4 for this geometric series. 2 The initial value represents our a value, so a = 2. 3 Before we can write the series in summation notation, we must determine the upper limit of the summation. 8192 = arn−1 = (2)(4n−1 ) 4096 = 4n−1 46 = 4n−1 6 = n −1 n = 7 Dr. Gabby (KNUST-Maths) Functions 34 / 41
32. 32. Sequence and Series So, we can rewrite the series as P6 k=0 2(4k ). From the formula for the sum of a geometric series, 6 X k=0 2(4k ) = 2 µ (4)7 −1 4−1 ¶ = 2(5461) = 10922 Therefore, the sum of the series, 2+8+32+128+...+8192 is 10922 Dr. Gabby (KNUST-Maths) Functions 35 / 41
33. 33. Sequence and Series Binomial Series In this final section of this chapter we are going to look at another series representation for a function. Before we do this let’s first recall the following theorem. Binomial Theorem If n is any positive integer then, (a +b)n = n X i=0 Ã n i ! an−i bi = an +nan−1 b + n (n −1) 2! an−2 b2 +···+nabn−1 +bn where, Ã n i ! = n (n −1)(n −2)···(n −i +1) i! ; i = 1,2,3,...n Ã n 0 ! = 1 Dr. Gabby (KNUST-Maths) Functions 36 / 41
34. 34. Sequence and Series Example (2x −3)4 = 4 X i=0 Ã 4 i ! (2x)4−i (−3)i = Ã 4 0 ! (2x)4 + Ã 4 1 ! (2x)3 (−3)+ Ã 4 2 ! (2x)2 (−3)2 + Ã 4 3 ! (2x)(−3)3 + Ã 4 4 ! (−3)4 = (2x)4 +4(2x)3 (−3)+ 3 2 (2x)2 (−3)2 +4(2x)(−3)3 +(−3)4 = 16x4 −96x3 +216x2 −216x +81 Dr. Gabby (KNUST-Maths) Functions 37 / 41
35. 35. Sequence and Series Binomial Series If k is any number and |x| < 1 then, (1+ x)k = ∞ X n=0 Ã k n ! xn = 1+kx + k (k −1) 2! x2 + k (k −1)(k −2) 3! x3 +··· where, Ã k n ! = k (k −1)(k −2)···(k −n +1) n! ; n = 1,2,3,... Ã k 0 ! = 1 Example Write down the first four terms in the binomial series for p 9− x Dr. Gabby (KNUST-Maths) Functions 38 / 41
36. 36. Sequence and Series We have to rewrite the terms into the form (1+ xk)k required, that is p 9− x = 3 ³ 1− x 9 ´1 2 = 3 ³ 1+ ³ − x 9 ´´1 2 So, k = 1 2 and xk = −(x/9) Then the binomial series is given, p 9− x = 3 ³ 1+ ³ − x 9 ´´1 2 = 3 ∞ X n=0 Ã 1 2 n ! ³ − x 9 ´n = 3 " 1+( 1 2 )(− x 9 )+ 1 2 (−1 2 ) 2 (− x 9 ) 2 + 1 2 (−1 2 )(−3 2 ) 6 (− x 9 ) 3 +··· # So the first four terms are = 3− x 6 − x2 216 − x3 3888 −··· Dr. Gabby (KNUST-Maths) Functions 39 / 41
37. 37. Sequence and Series Exercise 1 Find the period of the following functions 1) f (x) = sin(2x), 2) f (x) = cos(−2x +π/3), 3) f (x) = x −sin(x). 2 Find the domain of: 1)f (x) = 1 2x −6 2)f (x) = 1− x 1+ x 3)f (x) = x3 −2x x(−x −6) 4)f (x) = 3x −1− 1 2x −6 5)f (x) = x 1−2x + x2 6)f (x) = x2 −2x (x −3)(1− x2) 3 Determine whether the functions below are even, odd or neither. 1) f (x) = ex2 −1 +ln(|x|+1), 2) f (x) = x2 −2 x(1−x2) , 3) f (x) = x2 sin(x) 4) f (x) = x p |x|−1, 5) f (x) = ln ¡ tanx −e|x| ¢ , 6) f (x) = x −1. Dr. Gabby (KNUST-Maths) Functions 40 / 41
38. 38. END OF LECTURE THANK YOU Dr. Gabby (KNUST-Maths) Functions 41 / 41