d i E Functions a l l u d Dr. Abdulla Eid b A College of - PowerPoint PPT Presentation

Section 2.1 d i E Functions a l l u d Dr. Abdulla Eid b A College of Science . r D MATHS 103: Mathematics for Business I Dr. Abdulla Eid (University of Bahrain) Functions 1 / 26

d 1 Definition of a function. i E 2 Finding the domain of a function. a l l 3 Finding function values. u d 4 Application of functions. b A . r D Dr. Abdulla Eid (University of Bahrain) Functions 2 / 26

1. Definition of a function A function from a set X to a set Y is an assignment ( rule ) that tells how one element x in X is related to only one element y in Y . d i E Notation: a f : X → Y . l l u y = f ( x ) . ” f of x ”. d b x is called the input (independent variable) and y is called the output A (dependent variable). . r The set X is called the domain and Y is called the co–domain. While D the set of all outputs is called the range. Think about the function as a vending machine! Dr. Abdulla Eid (University of Bahrain) Functions 3 / 26

Question: How to describe a function mathematically? Answer: By using algebraic formula! Example Consider the function d f : ( − ∞ , ∞ ) → ( − ∞ , ∞ ) i E x �→ 3 x + 1 a l l u or simply by f ( x ) = 3 x + 1 d b A f(1)=3(1)+1=4. . f(0)=3(0)+1=1. r D f(-2)=3(-2)+1=-5. f(-7)=3(-7)+1=-20. Domain = ( − ∞ , ∞ ) . Co–domain= ( − ∞ , ∞ ) . Range= ( − ∞ , ∞ ) . Dr. Abdulla Eid (University of Bahrain) Functions 4 / 26

Example f : ( − ∞ , ∞ ) → ( − ∞ , ∞ ) x �→ x 2 d i or simply by f ( x ) = x 2 E a l f(1)= ( 1 ) 2 =1. l u d f(0)= ( 0 ) 2 =0. b f(-1)= ( − 1 ) 2 =1. A f(-2)= ( − 2 ) 2 =4. . r D f(14)= ( − 4 ) 2 =16. f(4)= ( 4 ) 2 =16. Domain = ( − ∞ , ∞ ) . Co–domain= ( − ∞ , ∞ ) . Range= [ 0, ∞ ) . Dr. Abdulla Eid (University of Bahrain) Functions 5 / 26

Example f : ( − ∞ , ∞ ) → ( − ∞ , ∞ ) x �→ 1 x d i E or simply by f ( x ) = 1 x a l l u f(1)= 1 1 =1. d f(-1)= 1 − 1 =-1. b A f(2)= 1 2 = 1 2 . − 4 = − 1 f(-4)= 1 . 4 . r D f(100)= 1 100 = 1 100 . f(0)= 1 0 =undefine (Problem, so we have to exclude it from the domain!) Domain = { x | x � = 0 } . Co–domain= ( − ∞ , ∞ ) . Range= { y | , y � = 0 } . Dr. Abdulla Eid (University of Bahrain) Functions 6 / 26

2. Finding the domain of functions Recall: The domain of f is the set of all x such that f ( x ) defined (makes sense). I.e., no problems like having a zero in the denominator or negative inside the square root, etc). so d i E Domain of f = { x | f ( x ) defined } a l l u d Example b A 3 (Zero denominator) Find the domain of f ( x ) = x − 1 . . r D Solution: Here we would have problems (undefined values) only if the denominator is equal to zero, so we need to find when the denominator is equal to zero and we exclude them from the domain. x − 1 = 0 → x = 1 So the domain of f is the set of all values except x = 1 Dr. Abdulla Eid (University of Bahrain) Functions 7 / 26

The domain 1- Set notation d Domain = { x | x � = 1 } i E � �� � your answer a l l u 2- Number Line notation d b A . r D 3- Interval notation ( − ∞ , 1 ) ∪ ( 1, ∞ ) Dr. Abdulla Eid (University of Bahrain) Functions 8 / 26

Exercise Find the domain of f ( x ) = 2 x + 1 3 x + 8 . d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Functions 9 / 26

Example x 2 − 1 (Zero denominator) Find the domain of f ( x ) = 3 x 2 − 5 x − 2 . d Solution: Similarly to the previous example, we would have problems i E (undefined values) only if the denominator is equal to zero, so we need to a find when the denominator is equal to zero and we exclude them from the l l u domain. d b A 3 x 2 − 5 x − 2 = 0 . r D x = − 1 x = 2 or ( Section 0.8 using the formula ) 3 So the domain of f is the set of all values except x = 2 and x = − 1 3 Dr. Abdulla Eid (University of Bahrain) Functions 10 / 26

The domain 1- Set notation Domain = { x | x � = 2 and x � = − 1 d 3 } i E a l l 2- Number Line notation u d b A . r 3- Interval notation D ( − ∞ , − 1 3 ) ∪ ( − 1 3, 2 ) ∪ ( 2, ∞ ) Dr. Abdulla Eid (University of Bahrain) Functions 11 / 26

Exercise 21 Find the domain of f ( x ) = x 2 + 3 . d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Functions 12 / 26

Example (Negative inside the root) Find the domain of f ( x ) = √ d 2 x − 4. i E a Solution: Here we would have problems (undefined values) only if there is l l a negative inside the square root, so we need to find all values that make u d 2 x − 4 is greater than or equal to zero, so we need to solve the inequality b A 2 x − 4 ≥ 0 → x ≥ 2 . r D So the domain of f is the set of all values x such that x ≥ 2 Dr. Abdulla Eid (University of Bahrain) Functions 13 / 26

The domain 1- Set notation d Domain = { x | x ≥ 2 } i E a l l 2- Number Line notation u d b A . r 3- Interval notation D [ 2, ∞ ) Dr. Abdulla Eid (University of Bahrain) Functions 14 / 26

Example (Negative inside the root and zero in the denominator) Find the domain of 3 f ( x ) = x − 4 . √ d i E Solution: Here we would have two problems (undefined values) only if there is a negative inside the square root or zero in the denominator, so we a l need to find all values that make x − 4 is is equal to zero and we exclude l u d them. Then we find all the values that make x − 4 non–negative, so we b need to solve the first A . r denominator = 0 and inside ≥ 0 D x − 4 = 0 and x − 4 ≥ 0 So the domain of f is the set of all values x such that x ≥ 4 and x � = 4 Dr. Abdulla Eid (University of Bahrain) Functions 15 / 26

The domain 1- Set notation d Domain = { x | x ≥ 4 and x � = 4 } i E a l l 2- Number Line notation u d b A . r 3- Interval notation D ( 4, ∞ ) Dr. Abdulla Eid (University of Bahrain) Functions 16 / 26

Exercise Find the domain of f ( x ) = 3 x 2 + 1 3 x + 6 . √ d i E a l l u d b A Exercise . r √ D x + 5 Find the domain of f ( x ) = 2 x − 4 . Dr. Abdulla Eid (University of Bahrain) Functions 17 / 26

Exercise (Old Exam Question) Find the domain of the following functions: f ( x ) = 2 x + 5. 4 g ( x ) = x 2 − 4 . h ( x ) = √ d 3 x + 1. i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Functions 18 / 26

Example 2 − √ x . � (Negative inside two roots) Find the domain of f ( x ) = d i Solution: Here we would have problems (undefined values) only if there is E a negative inside the square roots, so we need to find all values that make a l whatever inside the square root to be greater than or equal to zero, so we l u need to solve two inequalities at the same time d b 2 − √ x ≥ 0 A x ≥ 0 and . r D x ≥ 0 and 4 ≥ x So the domain of f is the set of all values x such that x ≥ 0 and 4 ≥ x . Dr. Abdulla Eid (University of Bahrain) Functions 19 / 26

The domain 1- Set notation d Domain = { x | x ≥ 0 and 4 ≥ x } i E a l l 2- Number Line notation u d b A . r 3- Interval notation D [ 0, 4 ] Dr. Abdulla Eid (University of Bahrain) Functions 20 / 26

Finding Function Values Recall ( a ± b ) 2 = a 2 ± 2 ab + b 2 d i E a Example l l u Let g ( x ) = x 2 − 2. Find d b f(2)= ( 2 ) 2 − 2=2. (we replace each x with 2). A f( u )= ( u ) 2 − 2= u 2 − 2. . r f( u 2 )= ( u 2 ) 2 − 2= u 4 − 2. D f( u + 1)= ( u + 1 ) 2 − 2= u 2 + 2 u + 1 − 2 = u 2 + 2 u − 1. Dr. Abdulla Eid (University of Bahrain) Functions 21 / 26

Exercise Let f ( x ) = x − 5 x 2 + 3 . Find f ( 5 ) . f ( 2 x ) . d f ( x + h ) . i E f ( − 7 ) . a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Functions 22 / 26

Example Let f ( x ) = x 2 + 2 x . Find f ( x + h ) − f ( x ) . h Solution: d i E = ( x + h ) 2 + 2 ( x + h ) − ( x 2 + 2 x ) f ( x + h ) − f ( x ) a l h l h u = x 2 + 2 xh + h 2 + 2 x + 2 h − x 2 − 2 x d b h A = 2 xh + h 2 + 2 h . r h D = h ( 2 x + h + 2 ) h = 2 x + h + 2 Dr. Abdulla Eid (University of Bahrain) Functions 23 / 26

Exercise Let f ( x ) = 2 x 2 − x + 1. Find f ( x ) − f ( 2 ) . x − 2 d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Functions 24 / 26

4. Application of Functions Example (Demand Function) The demand function is expressed as the following for certain item p = 120 d q , i E a where q is the number of units and p is the price for unit. If the price is 6 l l BD per unit, how many units we have? u d b Solution: A . p = 120 r D q 6 = 120 q 6 q = 120 q = 120 = 20 6 Dr. Abdulla Eid (University of Bahrain) Functions 25 / 26

Example d A company has capital of 7000 BD and weekly income of 320 BD and i E weekly expenses of 210 BD. Find the value V of the company in terms of a t which is the number of weeks. l l u Solution: d b V ( t ) = 7000 + ( 320 − 210 ) t A . r V ( t ) = 7000 + 110 t D Dr. Abdulla Eid (University of Bahrain) Functions 26 / 26