FUNCTIONS OBJECTIVES define functions distinguish between dependent and

FUNCTIONS

OBJECTIVES: • define functions; • distinguish between dependent and independent variables; • represent functions in different ways; and • evaluate functions • define domain and range of a function; and • determine the domain and range of a function

DEFINITION: FUNCTION • A function is a special relation such that every first element is paired to a unique second element. • It is a set of ordered pairs with no two pairs having the same first element. • A function is a correspondence from a set X of real numbers x to a set Y of real numbers y, where the number y is unique for a specific value of x.

Functions One-to-one and many-to-one functions Consider the following graphs and x maps to only one value of y. . . and each y is mapped from only one x. Each value of x maps to only one value of y. . . BUT many other x values map to that y. Each value of

Functions One-to-one and many-to-one functions Consider the following graphs and is an example of a one-to-one function is an example of a many-to-one function One-to-many is NOT a function. It is just a relation. Thus a function is a relation but not all relation is a function.

In order to have a function, there must be one value of the dependent variable (y) for each value of the independent variable (x). Or, there could also be two or more independent variables (x) for every dependent variable (y). These correspondences are called one-to-one correspondence and manyto-one correspondence, respectively. Therefore, a function is a set of ordered pairs of numbers (x, y) in which no two distinct ordered pairs have the same first number.

Ways of Expressing a function 1. Set notation 4. Graph 2. Tabular form 5. Mapping 3. Equation

Example: Express the function y = 2 x; x= 0, 1, 2, 3 in 5 ways. . 1. Set notation (a) S = { ( 0, 0) , ( 1, 2 ) , ( 2, 4 ), ( 3, 6) } or (b) S = { (x , y) / y = 2 x, x = 0, 1, 2, 3 } 2. Tabular form x y 0 0 1 2 2 4 3 6

3. Equation: y = 2 x 5. Mapping 4. Graph 5 4 3 2 1 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 y x y 0 1 0 ● ● ● 1 2 3 4 5 x 2 3 2 4 6

EXAMPLE: Determine whether or not each of the following sets represents a function: 1. A = {(-1, -1), (10, 0), (2, -3), (-4, -1)} 2. B = {(2, a), (2, -a), (2, 2 a), (3, a 2)} 3. C = {(a, b)| a and b are integers and a = b 2} 4. D = {(a, b)| a and b are positive integers and a = b 2} 5.

SOLUTIONS: 1. A is a function. There are more than one element as the first component of the ordered pair with the same second component namely (-1, -1) and (-4, -1), called a many-to-one correspondence. One-tomany correspondence is a not function but many-to-one correspondence is a function. 2. B is a not a function. There exists one-to-many correspondence namely, (2, a), (2, -a) and (2, 2 a).

3. C is not a function. There exists a one-to-many correspondence in C such as (1, 1) and (1, -1), (4, 2) and (4, -2), (9, 3) and (9, -3), etc. 4. D is a function. The ordered pairs with negative values in solution c above are no longer elements of C since a and b are given as positive integers. Therefore, one-tomany correspondence does not exist anymore in set D. 5. E is not a function Because for every value of x, y will have two values.

OTHER EXAMPLES: Determine whether or not each of the following sets represents a function: a) b) c) d) e) f) S = { ( 4, 7 ), ( 5, 8 ), ( 6, 9 ), ( 7, 10 ), ( 8, 11 ) } S = { ( x , y ) s. t. y = | x | ; x R } y=x 2 – 5 |y|= x

DEFINITION: FUNCTION NOTATION • Letters like f , g , h, F, G, H and the likes are used to designate functions. • When we use f as a function, then for each x in the domain of f , f ( x ) denotes the image of x under f. • The notation f ( x ) is read as “ f of x ”.

EXAMPLE: Evaluate each function value. 1. If f ( x ) = x + 9 , what is the value of f ( x 2 ) ? 2. If g ( x ) = 2 x – 12 , what is the value of g (– 2 )? 3. If h ( x ) = x 2 + 5 , find h ( x + 1 ). 4. If f(x) = x – 2 and g(x) = 2 x 2 – 3 x – 5 , Find: a) f(g(x)) b) g(f(x)) 5. If find each of the following

6. Find (a) g(2 + h), (b) g(x + h), (c). . where h 0 if 7. Given that show that 8. If 9. If , show that find

DEFINITION: Domain and Range All the possible values of x is called the domain and all the possible values of y is called the range. In a set of ordered pairs, the set of first elements and second elements of ordered pairs is the domain and range, respectively. Example: Identify the domain and range of the following functions. 1. ) S = { ( 4, 7 ), ( 5, 8 ), ( 6, 9 ), ( 7, 10 ), ( 8, 11 ) } Answer : D: { 4, 5, 6, 7, 8} R: {7, 8, 9, 10, 11}

2. ) S = { ( x , y ) s. t. y = | x | ; x R } Answer: 3) D: all real nos. R: all real nos. > 0 y=x 2 – 5 Answer. D: all real nos. R: all real nos. > -5

4. Answer: D: all real nos. except -2 R: all real nos. except 2 5. Answer : D: all real nos. > – 1 R: all real nos. > 0 6. Answer: D: all real nos. <3 R: all real nos. <0

From the above examples, you can draw conclusions and formulate the following theorems on the domain determination of functions. Theorem 1. The domain of a polynomial function is the set of all real numbers or (- , + ). Theorem 2. The domain of is the set of all real numbers satisfying the inequality f(x) 0 if n is even integer and the set of all real numbers if n is odd integer. Theorem 3. A rational function f is a ratio of two polynomials: The domain of a rational function consists of all values of x such that the denominator is not equal to zero where P and Q are polynomials.

KINDS OF FUNCTIONS: 1. An algebraic function is the result when the constant function, (f(x) = k, k is constant) and the identity function (g(x) = x) are put together by using a combination of any four operations, that is, addition, subtraction, multiplication, division, and raising to powers and extraction of roots. Example: f(x) = 5 x – 4,

Generally, functions which are not classified as algebraic function are considered as transcendental functions namely the exponential, logarithmic, trigonometric, inverse trigonometric, hyperbolic and inverse hyperbolic functions.

EXERCISES: A. Which of the following represents a function? 1. A = {(2, -3), (1, 0), (0, 0), (-1, -1)} 2. B = {(a, b)|b = ea} 3. C = {(x, y)| y = 2 x + 1} 4. 5. E = {(x, y)|y = (x -1)2 + 2} 6. F = {(x, y)|x = (y+1)3 – 2} 7. G = {(x, y)|x 2 + y 2 = } 8. H = {(x, y)|x y} 9. I = {(x, y)| |x| + |y| = 1} 10. J = {(x, y)|x is positive integer and

B. Given the function f defined by f(x) = 2 x 2 + 3 x – 1, find: a. f(0) f. f(3 – x 2) b. f(1/2) g. f(2 x 3) c. f(-3) h. f(x) + f(h) d. f(k + 1) i. [f(x)]2 – [f(2)]2 e. f(h – 1) j.

C. Given find

EXERCISES: Find the domain and range of the following functions:

Exercises: Identify the domain and range of the following functions. 1. {(x, y) | y = x 2 – 4 } 6. y = | x – 7 | 2. 7. y = 25 – x 2 3. 8. y = (x 2 – 3) 2 4. 5. 9. 10.