A One to One function
A function is one to one if no more
than one element in the domain is
paired with an element in the range.
Examples of one to one
functions
1. A student and his or her spouse
2. A student and the college at which he
or she is taking precalculus
3. A person and his or her social security
number
4. A person and his or her passport
number
5. Any function whose graph passes the
horizontal test
Examples of one to one
functions
Note every Uppercase number has
exactly one lower case partner
A
A
B
B
C
C
a
b
c
Examples of one to one
functions
1.
f ( x)  x
3
2. f (x) = 3x +5
3. f ( x) 
x
4. All odd functions
Examples of functions that
are not one to one
1. f (x) = |x|
2. f ( x)  x 2
3. f ( x ) 
1
x2
Examples
 Two function are inverse functions if the domain of one
is the range of the other. 3x and x/3 are inverse
function
Domain Range
x
3x
Domain x
Range
x/3
-6
-2
-3
-1
0
0
3
1
-2
-6
-1
-3
0
0
1
3
6
2
2
6
9
3
3
9
The inverse of a function
1
1
f is written
fof ( x)  f ( f ( x))  x
f
1
Inverse Functions
 Are f ( x)  2 x  6
 2(1 x - 3) +6
2
x–6+6=x
1 (2x + 6) – 3
2
 x + 3 – 3= x
and
1
g ( x)  x  3
2
Graphs of Inverse Functions
f ( x)  x 3
f ( x)  3 x
f ( x)  x 3
Note inverse functions are reflected about the
line y = x
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A One to One Function