Properties of Functions Section 4.3 Properties of Functions

Activity 1 Define One to one functions Onto functions Invjective functions Surjective functions Bijective functions

Definition We can symbolically define that a function is one-to-one (injective) with: f: XY is one-to-one  x1x2 f(x1) = f(x2)  x1 = x2

Definition f: XY is one-to-one  x1x2 f(x1) = f(x2)  x1 = x2 From this statement, how can we determine the definition of a function that is not a one-to-one function? Take the negation. DO IT x1x2 f(x1) = f(x2)  x1  x2

Equivalent Definitions (contrapositive) Mine: f: XY is one-to-one  x1x2 f(x1) = f(x2)  x1 = x2 Your book’s: x1x2 x1  x2  f(x1)  f(x2)

Activity 2 Here is a function. f: RR f(x) = 2x + 3 Is it a one-to-one function? Could you prove it ?

Activity 2 To be one-to-one we must show that x1x2 f(x1) = f(x2)  x1 = x2 [definition of one-to-one function] Suppose x1 and x2 are real numbers such that f(x1) = f(x2) 2x1 + 3 = 2x2 + 3 substitution of definition of f(x) 2x1 = 2x2 Algebra x1 = x2 Algebra  f(x) is a one-to-one function

Activity 3 Here is a function. f: RR f(x) = x2 + 4 Is it a one-to-one function? Could you prove it ?

Activity 3 Use a Counter example Let x1 = 2 and x2 = -2 f(x1) = 22 + 4 = 8 f(x2) = (-2)2 + 4 = 8 Different values of x give same value of f(x)

Definition We can symbolically define that a function is onto (surjective) with: f: XY is onto  yY xX f(x) = y

Definition How could we define that a function is NOT onto (surjective) ? f: XY is not onto  yY xX f(x)  y

Activity 4 Return to the function. f: RR f(x) = 2x + 3 Is it an onto function? Could you prove it ?

Activity 4 Suppose that y is a particular but arbitrary real number. We need to show that there is some real number x whose image is y. If such a real number exists, then 2x + 3 = y definition of our function x = (y - 3)/2 algebra x is a real number closure  f(x) = 2x + 3 is an onto function

Activity 5 Return to the function. f: RR f(x) = x2 + 4 Is it an onto function? Could you prove it ?

Activity 5 We need to find a counterexample. That is, we need to find a value in the target domain such that yY xX f(x)  y y = -1 No value of x plugged into x2 + 4 gives value of of -1