Functions: Definition A function relate an input to output In mathematics, a function is a relation between a set of outputs and a set of output with the property that each input is related to exactly one output. of all the x for A function f: X to Y is a rule that assigns, to each element x of X, at most one element of Y. If an element is assigned to x in X, it is donated by f(x). The subset of X consisting which f(x) is defined is called the Domain of f.the set of all element in Y of the form of f(x), is called the range of f General form for a function in many independence variables Y=G(X i ), i = 1,2,3,,n Where :Y: is dependent variable, X: are independent variables. An example Defines y as a function of x. The equation gives the rule add 2 to the value of x Which means when x=2, y=4 and when x = -6 then become y=-4
F(x), which is read f of x and which means the output, in the range of f, that results when the rule f is applied to the input x, from the domain of f. The outputs is called Function values. Equality of Functions: To say that two functions f and g in terms of x and y are equal, denoted g=f,is to say that : The domain of f is equal to the domain of g. For every x in the domain of f and g.f(x)=g(x). Example : Determine whether the following functions are equal. f(x)=x+2 g(x)=(x+2)(x-1) (x-1) It is easy to find that f and g are equal
Exercise(1) Determine whether the given functions are equal. 1. F(x)=x, g(x)=. 2. H(x)=x+1, f(x)=( ) 2. Find the indicated value for the given function: 1. f(x)=2x+1, f(0), f(3),f(-4). 2. f(u)=2u 2 -u,f(-2), f(2v),f(x+a). 3. h(x)=(x+4) 2,h(0), h(2),h(t-4).
Special functions : Term special concerns about functions having special forms and representations, and we will begin with the simplest type, which is called constant function. Constant function we call h a constant function because all the function values are the same, in a more specific way it is a function of the form: h(x) = c, where c is a constant, is called a constant function. polynomial function A constant function belongs to a boarder class of functions, called polynomial function. In general, a function of the form f(x)=qx n +wx n-1 +ex n-2 +..+c. n is a positive integer,where q, w, e and c are constants. It is called polynomial function, n is the degree of a polynomial And q is the leading coefficient.
polynomial function can be function in one dependent variable or more than one variable. The Linear function is a function of degree one, take the form: The square polynomial function : Is polynomial function of the degree 2. Cubic function Is Polynomial function of degree 3. It is take the form : n i x a x f y x a x a x a a x f y i i a 1,2,3,.., ) ( ) ( 3 3 2 2 0 n i x a x f y x a x a x a x a a x f i i i i n n n i 1,2,3,...,... ) ( 3 3 3 2 2 2 1 1 0
Examples of a polynomial functions Is a polynomial function of degree 3 with leading coefficient 1 Is a linear function with leading coefficient 2 3 Rational Functions A rational function is formed by dividing one polynomial by another polynomial, for example: f(x)=, Note that the polynomial function is a rational function but the denominator equals 1 Example :.
Exercise (2) 1/ Determine whether the given functions is a polynomial function : f(x) = X 2 x 4 +4 g(x)=7 x+4 j(x) = 2-3x 2/ Determine whether the given functions is a rational function : f(x) =(x 2 +x) (x 3 +4) g(x)= 4x -4 m(x)= 3 2x+1 3/ state the degree and the leading coefficient of the given polynomial function : g(x)= 9x 2 +2x +1 f(x)=1 π -3x 5 +2x 6 + x 7 e(x)= 9 4/ find the function values for each function: g(x) = 3- x 2, g(10), g(3), g(-3) f(x)=8, f(2), f(t+8)
Combinations of Functions : There are many ways of combining two functions to create a new one. suppose f and g are the functions given by : f(x)=7x g(x)=2x 2 by adding : h(x)=7x+2x 2 by subtracting : r(x)= 7x-2x 2 or :e(x)= 2x 2-7x by multiplying: t(x)=14x 3. By dividing: c(x)=0.3x Note : for each new function the resultant domain is set of all x which belong to both the domain of f and g Also : (cf)(x)= c.f(x)
Exercise (3) 1. If f(x)= 3x-1, g(x)=x 2 + 3x, find : 2. (f+g)(x) 3. (f-g)(x) 4. (fg)(x) 5. (f g) (x) 6. (0.5f)(x)
Exponential Functions : The functions of the form f(x) = b x, for constant b, are important in mathematics, business, economics, science and other areas of study. An example is f(x) = 2 x. such functions are called exponential functions. Definition The function f defined by : F(x) = b x Where b 0, b 1, and the exponent x is any real number, is called an Exponential Functions with base b. Rules for exponents : b x b y = b x+y (b x ) y = b xy (b c) x = b x c x ( )
b 1 = b b 0 = 1 (b. c -1 ) x = b x c -x Example 1 : Bacteria Growth: The number of bacteria present in a culture after t minutes is given by : N(t) = 300 (4 3) t How many bacteria are present initially? Solution: Here we want to find N(t) when t =0. we have: ( ) Thus, 300 bacteria are initially present. Approximately how many bacteria are present after 4 minits?
Exponential Function with base e : The number e provides the most important base for an exponential function. In fact the Exponential function with base e is called the natural exponential function and even the exponential function to stress it is importance. It has a remarkable property in calculus. it also occurs in economic analysis and problem involving exponential growth, Logarithmic Functions Y= e x Y= log b x if and only b y =x And we have: log b b x = x (1) b log b x = x..(2) Where equation (1) holds for all x in (-, + ) the domain of the exponential function with base b And equation (2) holds for all x in (0, ) the range of the exponential function with base b
Logarithmic Function properties 1- Log b (mn) = log b m + log b n. And log b m + log b n = log b (mn). For example : Log 56 = log (8. 7) = log 8 + log 7 2- Log b (m n) = log b m log b n For example : Log (9 2) = log 9 log 2 3- Log b m r = r log b m Example : log 64 = log 8 2 = 2 log 2 3 = 2 * 3 log 2 = 6 log 2. 4- Log (1 m) = - log m Example : Log ¼ = -log 4 log(2 3) = - log (3 2)
Piecewise Defined Function Let F(x)= -2 x if x -3 3x-1 If -3 x 2-4x if x 2 This is called Piecewise Defined Function, because the rule for specifying it is given by rules for each of several disjoin case. Where s is the independent variable, and the domain of F is all (S) such that F(-5)= -2 (-5) = 10 (-5, 10) F (-1) = 3 (-1) -1= -4 (-1,-4) F (-3) = 3(-3) -1= -10 (-3,-10) F (4)= -4x =-4 (4)= -16 (4,-16) Piecewise Defined Function: e.g F(x) = x-2 if x <3 5-x if x 3
F (-5)= x-2= -5-2=07 F (-1)= x-2= -1-2=-3 F (0)= x-2= 0-2=-2 F (3)= 5-x= 5-3=2 F (5)= 5-5=0 Absolute value function: The function -1 (x)= 1 is called absolute value function. Recall that absolute value of real number x is a function denoted by x and defined by x = x if x 0 -x if x 0 Thus the domain of - is the real numbers. Some functions value are 16 = 16-4/3 = - -4/3 = 4/3
0 = 0 = e.g f (x)= 2x + 6 = then f (3)= 2(3) +6 = 12 = 12 f (-4)= 2 (-4) + 6 = -2 = 2 Inverse Function: Just as -a is the number for which a + -a =0= -a + a for a 0, a -1 is the number for which a a -1 = 1 = a -1 a In mathematical, g, is uniquely determined by f and is therefore given name, g=f -1 is real as f inverse and called the inverse function of f. To get the inverse of function by doing the following 1- replace f (x) = y 2- Interchange x & y. 3- Solve for y 4- Replace y with f -1
Examples: F (X)= (X-1) 2 Find the F -1 Solution: let y= (X-1) 2 Example: F(x)= x+4 Find F -1
Example: f(x)= x 3 +3, Find F -1 Example: f (x)= 3x+2 Find F -1
Exercise(4) Find The inverse of F(x)= 3x+7 G (x) = 5x-3 F (x) = (4x-5) 2
Finding domains Find the domain of the each function. Solution : we cannot divide by zero, so we must find any values of x that make the denominator = 0. these cannot be inputs. Thus, we set the denominator equal to 0 and solve for x : after factoring: so: ( then the domain is all real numbers except -1 and 2. Find the domain of the function: solution :
the domain is a closed interval starting from 0.5 to infinite (). Example: The domain s all the real number except -2, -3 Find the domain of the function : the domain the real numbers, {, -3} U { -2, }
Exercise(5)
Equations Linear Equations: Definition: A linear Equation in the variable (x) is an equation that is equivelant to one that can be written in the form: Where:, a and b are constants, and a 0 A linear equation is also called a firt-degree equation or an equation of the degree one since the highest power of the variables is (1) Solving a linear equation:,eg : solve the following;
Solve : Solving a linear Equation: Multiply both sides by (4): ( )
Quadratic Equation: The Quadratic Equation in the variable X is an equation that can be written in the form where a, b &c are constant, a 0 Solution by factoring: the solution set is (3, -4). Solve: W = 0, or 6w = 5 w= 5/6 the solution set is (0, 5/6). (3x-4)(x+1)= -2 We first multiply the factors on the left side
4x- 4x 3 = 0 X 2 +2x-8=0 X 2 +2x = 8 by adding 1 to the both side:
Exercise(6) Solve the following by factoring: 1-2- 3-4-
Quadratic formula Solve By using quadratic formula Solution:
Exercise(7) Solve the following by using Quadratic formula 1-2- 3-
Systems of Linear Equations Two variable Systems : Any set of two linear equations is called a set of equations in the variable x and y or any other variables. The main concern in this section is algebraic methods of solving a system of linear equations. We will successively replace the system by other systems that have the same solutions. We say that equivalent systems of equations. The replacements systems have progressively more desirable form of determining the solution. Our passage from a system to an equivalent system will always be accomplished by one of the following procedures : 1- Interchanging two equations. 2- Multiplying one equation by a nonzero constant. 3- Replacing an equation by itself plus a multiple of another equation.
Example: consider two equations : Solution : Multiply equation (1) by 9 and equation (2) by -4, the resulted equations are : Then by summing the two equations : So y = 35, then by putting y =35 in equation (1) then we find that x = 40. We can check our answers by substituting x= 40 and y =35 into both of the ongoing equations. if the answers are the same, then our solution is true.
this method is called the addition method Example: Choose one of the equations, for example equation (1), and solve it for one variable in terms of the other, say x in terms of y,then substitute it in the other equation : From (1) : In (2) : substitute x, then it will be in the form : Then y = -1 So x = 5,
then investigate whether the solution is correct or not. In (1) : 5-3 = 2 In (2) : 10+ 4 = 14 (correct) (correct) This method is called the elimination by substitution
Exercise(8) Solve the systems algebraically : 1-2- 3-