Section 4.3
Properties of Functions
Activity 1
• Define
1.
2.
3.
4.
5.
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 oneto-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:
f: XY is one-to-one 
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