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Real Functions Real Functions September 25, 2011 1 Introduction A function over D, being D is a rule that for every number x of D ⊂ ℜ associates only one number of . It is represented in the form f : D . ℜ → ℜ y is called image of x and it is written as y = f(x). It is understood by domain of the function f the set: Domf = x : y with y = f(x) { ∈ℜ ∃ ∈ℜ } While the image of the function is defined as the set: Imf = y : x with y = f(x) { ∈ℜ ∃ ∈ℜ } The graphic of the function f is defined as the subset of 2: ℜ G(f)= (x,y) 2 : y = f(x) { ∈ℜ } 2 Operations with functions Given two functions f, g : , we define the following operations: ℜ←ℜ sum of two functions: • (f + g)(x)= f(x)+ g(x) The domain of f + g is the intersection of the domains Dom f Dom g. ∩ product of two functions: • (f g)(x)= f(x) g(x) · · The domain of f + g is the intersection of the domains Dom f Dom g. ∩ Composition of two functions: • (g f)(x)= g(f(x)) ◦ This function exists for the x so that f(x) Dom g, for every x ∈ ℜ ∈ ∈ ℜ so that Imf Dom g. ⊂ 1 2.1 Kinds of functions Given a function f : , it is said that it is an even function when it • ℜ→ℜ verifies that: f( x)= f(x), x Domf − ∀ ∈ . This function would present symmetry with respect to OY axis. and it is odd when it verifies that f( x)= f(x), x Domf − − ∀ ∈ . This function would present symmetry with respect to the origin. Given a function f : , it is said to be periodic if it verifies that • ℜ→ℜ it exists a real number T that f(x + T )= f(x), x Dom f ∀ ∈ The minimum number T that verifies the previous condition is called period of the function. 3 Elemental functions A constant function has the form f(x)= c, being c any real number. This • functions transforms every real number in the number c. Its domain is ℜ and the image is c. A lineal function has the form f(x) = mx, being m a real constant • different from zero. They verify that f(αx1 + βx2) = αf(x1) + βf(x2). This function has as domain and image every real number. Its graphic is a straight line that crosses the origin with a slope m. If m> 0 the function is strictly increasing. If m< 0 the function is strictly decreasing. Affine functions have the form f(x) = mx + n, where m,n and • ∈ ℜ m = 0. This functions have as domain and image all the real numbers. Its 6 graphic is a straight line of equation y = mx + n, that crosses the point (0,n) with a slope m. Potential functions have the form f(x) = xn, where n . This • ∈ ℜ functions have as domain all the real numbers. Their image depends on the parity of n. If n is even, the image is all positive real numbers [; + inf). If n is odd, then the image is every number in . ℜ 2 Dirichlet function. It is defined as follows: • 0 ifx Q, f(x)= ∈ (1) 1 ifx/ Q. ∈ Its domain is every real number, rationals and irrationals. Its image is composed only by the numbers 0, 1 . Every rational has as image 0 and { } every irrational has as image 1. The sign function. It is defined as follows: • 1 if x> 0, sign(x)= 0 if x = 0, (2) 1 if x< 0. − Its domain is every real number. Its image is composed only by the numbers 0, 1, 1 { − } The absolute value function. It is defined as follows: • x ifx> 0, x = (3) | | x ifx< 0. − 3.1 Conic functions Circumference. The equation of a circumference with centre at (0,0) and • radius r os x2 + y2 = r2. Isolating y, y = r2 x2 ±p − that is not a function (for one value of x we get two values of y). We can consider two functions instead, y = r2 x2 p − y = r2 x2 −p − Their domain is [-r;r] and their images are [0;r] and [-r;0] respectively. Ellipse. The equation of an ellipse with centre at (0,0) and semi-axes • x2 y2 a, b > 0 a2 + b2 = 1. Isolating y, b y = a2 x2 ±ap − that is not a function. We can consider two functions instead, 3 b y = a2 x2 ap − b y = a2 x2 −ap − Their domain is [-a;a] and their images are [0;b] and [-b;0] respectively. Hyperbola. The equation of an ellipse with centre at (0,0) and semi-axes • x2 y2 a, b > 0 2 2 = 1. Isolating y, a − b b y = x2 a2 ±ap − that is not a function. We can consider two functions instead, b y = x2 a2 ap − b y = x2 a2 −ap − Their domain is ( ; a] [a; ) and their images are [0; ) ( ; 0] −∞ ∪ ∞ ∞ ∪ −∞ respectively. Parabola. The most used parabolas are those that have as equations • y = ax2 +bx+c with a, b, c (their domain is ) and y2 = px with p . ∈ℜ ℜ ∈ℜ Isolating y, b y = √px ±a that is not a function. We can consider two functions instead, b y = √px a b y = √px −a The domain for these two functions is [0; + ) if p > 0 and ( ; 0] if ∞ −∞ p< 0. 3.2 Trigonometric functions Sine, Cosine and Tangent. This functions are periodic and not inyective • so we have to restrict their domain to an interval where they are inyective. For function y = sin(x) we consider the interval [ π ; π ] and its image is − 2 2 [ 1; 1]. Its inverse function is y = arcsin(x), its domain is the image of − y = sin(x) and is image is [ π ; π ]. For y = cos(x) we choose the interval − 2 2 4 [0; π] where it is inyective and strictly decreasing. Its image is [ 1; 1]. Its − inverse function is the function y = arccos(x) with domain [ 1; 1] and image − [0; π] and also strictly decreasing. For the function y = tg(x) we consider the interval [ π ; π ] (the same as the sin function). Where it is inyective and − 2 2 increasing. Its inverse function y = arctg(x) transforms into this interval. ℜ Trigonometric functions verify important and fundamental relations: cos2(x)+ sin2(x) = 1 sin(x) tg(x)= cos(x) 1 1+ tg2(x)= cos2(x) Relations of addition: sin(x y) = sin(x) cos(y) cos(x) sin(y) ± ± cos(x y) = cos(x) cos(y) sin(x) sin(y) ± ∓ tg(x) tg(y) tg(x y)= ± ± 1 tg(x)tg(y) ∓ Relations of double arc: sin(2x) = 2sin(x) cos(x) cos(2x) = cos2(x) sin2(x) − Relations of half arc: x 1 cos(x) sin( )= − 2 ±r 2 x 1 + cos(x) cos( )= 2 ±r 2 Reciprocal trigonometric functions: Secant. • 1 sec(x)= cos(x) Cosecant. • 1 cosec(x)= sin(x) Cotangent. • 1 cotg(x)= tg(x) 5 3.3 Exponential functions The exponential function of base a is defined as f(x) = ax with a > 0. In the case a = 1 then the function is constant and equal to 1 f(x) = 1. Its graphic is different if 0 <a< 1 or a> 1. 1 n The number e is the limit of the series (1 + n ) , 1 e = lim(1 + )n n n 3.4 Logarithmic functions The logarithmic function of base a, being a > 0 and a = 1, is given by 6 y = loga(x). Exponential and logarithmic functions are inverse functions. The most important properties of the logarithms are the following: loga(xy)= loga(x)+ loga(y) x log ( )= log (x) log (y) a y a − a n loga(x )= nloga(x) When a is not specified a = 10 and when a = e we write y = ln(x) and we call it Neperian Logarithm. 3.5 Hyperbolic functions Hyperbolic functions are based on the exponential function: ex e−x sh(x)= − 2 ex + e−x ch(x)= 2 ex e−x th(x)= − ex + e−x These functions verify the relations: ch(x)2 sh(x)2 = 1 − sh(x) th(x)= ch(x) 1 1 th(x)2 = − ch(x)2 The argument functions of hyperbolic functions are also interesting: argsh(x), argch(x), argtgh(x) 6 4 Representation of functions 4.1 Asymptotes An asymptote is a straight line such that the distance between the curve and the line approaches zero as they tend to one specific value. They can be horizontal, vertical or obliques. The line of equation x = a is a vertical asymptote of the function f if lim f(x)= inf when x a, when x a+ or when x a−. It means that ± → → → when the function f approximates to the point a, at least from one lateral, the values of the function goes to inf. ± The straight line of equation y = k is an horizontal asymptote of function f if lim f(x)= k, or lim f(x)= k with k x→+∞ x→−∞ ∈ℜ This means that the distance between the asymptote and the function goes to zeros as x goes to infinity. There can be two horizontal asymptotes, one for + and another for , that can be the same line, there can ∞ −∞ exist an asymptote for only one of this values or there can be no horizontal asymptotes. The straight line of equation y = mx + n is said to be an oblique asymp- tote of the function f if x f(x) y 0, →±∞⇒ − → what is to say, the distance between the function and the asymptote goes to zero when x goes to plus or minus infinity.
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