College of Engineering MATHEMATICS I Common / Standard Functions Dr Fuad M. Shareef 1

Relations / Mappings Let A and B be two sets. A binary relation, R, from A to B assigns to each ordered pair (a,b) exactly one of the following statements: (i) “a is related to b”, written as aRb or (a,b)  R, (ii) “a is not related to b", written as a b or (a,b) R. Binary relation can be thought of as a statement which, given any two elements of a set A ( an ordered pair ), is either true or false for that pair. We know the relation completely if we know the set of pairs for which it is true. 2

Example: Let A ={1,2,3}, and R={(1,2),(1,3),(3,2)}, then R is a relation on A since it is a subset of AxA. With respect to this relation: 1R2, 1R3, 3R2 But the following are not 1 1, 2 1, 2 2, 2 3, 3 1, 3 3. The domain of R is {1,3} The range of R is {2,3}. 3

Domain and Range Domain of a function: consists of all values which the rule may be applied. The set of values that are valid in the rule The first coordinates of the ordered pairs When we write a function, both the rule and the domain is given. If the domain is not given, then it is assumed to be defined for all values that are valid in the rule. 4

Domain of functions Given Find: f(2) f(3) f(0) the least possible and the largest possible domain for f Domain is : 5

Domain of functions Given Find: f(2) f(-3) f(0) f(-5) the least possible and the largest possible domain for f Domain is : 6

Range of functions Range of a function: The set of values produced from applying the rule to the domain of the function. The set of out put values of the function Example1: For The range R is: Example2: For The range R is: 7

Graph of common functions Linear function 8 f(x)=2x+3 f(x)= -2x-3

Graph of common functions Quadratic function- Parabola 9 f 1 =x 2 f 2 = - x 2

Graph of common functions Cubic function 10

Graph of common functions Surd function –square, cubic, fourth (roots) 11

Exponential & Logarithmic Functions Exponential Function: Where f is called an exponential function. 12 Y=f(x)=2 x Y=3 x f(x)=4 x Y=f(x)=(0.5) x

Exponential & Logarithmic Functions Logarithmic Function: Or 13

Rational Functions Rational Function: Simple example: More complicated: 14 The lines x=0 and x=1 are called Asymptotes.

Exercise: Complete the graph of in the interval (0,1) 15 The lines x=0 and x=1 are called Asymptotes

Trigonometric Functions Sine Function: Range: 16

Trigonometric Functions Cosine Function: Range: 17

Trigonometric Functions Tangent Function: Range: 18

Tutorial Exercise, Assignment and Lecture notes Visit: the courses website once everyday. 19