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Unit 2 Section 3

# Unit 2 Section 3 - 2.3 PROPERTIES OF FUNCTIONS A FUNCTIONS...

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1 2.3 PROPERTIES OF FUNCTIONS A. FUNCTIONS CLASSIFIED BY ASSOCIATION Functions can also be classified not just by its algebraic type but also in terms of association such as 1-1, onto or bijective. Definition 2.3.1 . One to One Function A function f is said to be one-to-one(1-1) if and only if whenever a and b are two numbers in the domain of f and a b , then   f a f b . Equivalently, f is one-to-one iff whenever f(a) = f(b), then a b . No element of B is the image of more than one element in A . In a one-to-one function, given any y there is only one x that can be paired with the given y . Such functions are also referred to as injective . Graphically, a function f is one-to-one if and only if every horizontal line intersects the graph of the function in at most one point. This is called the horizontal line test . Illustration 2.3.1 Figure 2.3.1 Figure 2.3.2 Figure 2.3.3 x y x y y x "One-to-One" a b c d 4 2 1 5 8 A B A B a b c d 4 2 5 NOT "One-to-One"

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2 By the horizontal line test , the functions with graphs appearing in Figure 2.3.1 and Figure 2.3.2 are both one-to-one since any horizontal line will intersect their graphs in no more than one point. However, the function with graph appearing on Figure 2.3 .3 is not one-to-one since if a horizontal line is drawn anywhere (except along the origin or below the x -axis), it will intersect the graph in more than one point. Example 2.3.1 Determine which of the given functions is one to one. a. 7 3 ) ( x x f b. x x x g 1 2 ) ( 2 ) ( . 2 x x x h c Solution: a. Let a and b be elements of the domain and assume that f(a) = f(b). Then 3a 7 = 3b 7. Then it follows that a = b. Hence f is 1-1. b. The given function g is rational with both linear in the numerator and denominator, the graph in Figure 2.3.4 shows that if we draw a horizontal line we can’t find two ordered pairs having the same second component. Thus g is a one to one function. Equivalently, the function g can be shown to be 1-1 in the following manner: Let a and b be elements in the domain of g and let g(a) = g(b). Then, Figure 2.3.4
3 b 1 b 2 a 1 a 2 . Then 2ab b = 2ab a. It follows that a = b. c. The function h is quadratic and the graph is a parabola going upward as shown in Figure 2.3.5, hence it is not one to one since when can get two values of x, for a particular y, example y =0, when x =2 or x = -1. Equivalently, to show that h is not 1-1, let y = 0. Then x 2 x 2 = 0 implies x = 2 or x = -1. Thus f(2) = f(- 1) but 2 ≠ -1. Definition 2.3.2 . Onto Functions A function f from A to B is called onto if for all b in B there is an a in A such that f ( a ) = b . That is, each element in B has a corresponding partner (or pre-image) in A under f . Such functions are also referred to as surjective . Similarly, if f is a function from A to B then f is onto iff Rng(f) = B.

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