Functions Definition
A function from a set S to a set T is a rule that assigns to each
element of S a unique element of T . We write f : S → T .
Example
Let S = R, T = R, and let f  x   x 2 . This is mathematical
shorthand for the rule “assign to each x S its square.”
Determine whether f : R → R Is a function.
Solution
We see that f is a function since it assigns to each element of S a
unique element of T — namely its square.
Note
A nice, geometrical way to think about the condition that
Each x in the domain has corresponding to it precisely one
y value is this:
graph of function
not function
If every vertical line drawn through a curve intersects that
curve just once, then the curve is the graph of a function.
The Domain and Range
The set S
(called the domain of the function)
The set T
(called the range of the function)
x
(input)
f
(output)
Examples
Determine the domain of:
1
1 f  x  
x 2
Domain is: R  2
 2
f x   x  3
Domain is: x  3  0
x  3
3, 
Example
Let f  x   1  x 2 . Determine the domain of f.
Solution
----
++++
-1
Domain is:
----1
1,1
Example
Let f  x   x 2  3x  2 . Determine the domain of f.
Solution
+++++
-----1
Domain is:
+++++
2
 ,1
 2,  
Note
Domain is the projection of the graph on
x-axis
Range
is the projection of the graph on
y-axis
Example
Let f  x   x . Determine a domain and range for f
which make f a function.
y-axis
x-axis
Example
The graph of a function f is shown in figure.
(i) Find the value of f 1 and the zeros of the function.
(ii) What are the domain and range of f.
Solution
f 1  4
3  x  6
zero is at
x 6
2  y  4
Composition of Functions
* Suppose that f and g are functions
* The domain of g contains the range of f .
* If x is in the domain of f then g may be applied to f (x).
This is called g composed with f
or
the composition of g with f
g
f  x   g  f  x  
Example
Let f  x   x2 1 and g  x   3x  4 .
Calculate g ◦ f and f ◦ g .
Solution
g
f
 x   g  f
 g x
(x ) 
2
f
g  x   f  g (x ) 
 1
 f  3x  4 
 3  x 2  1  4
  3x  4   1
 3x 2  1
 9 x 2  24 x  15
g
f
 x    f
2
g  x 
The Inverse of a Function
Let f : S  T be a function. We say that f has an inverse if
f
there is a function
such that
f
f
f f
1
1
:T  S
x   x
1
x   x
Notice that the symbol f
we call the inverse of f .
1
denotes a new function which
Example
Find the inverse of the function f  x   3 x  1
Solution
We solve the equation
ff
1
3 f
 x   x
1
 1  x
3f
f
1
1
  x 1
x 1

3
Example
Find the inverse of the function f  x   2 x5
Solution
We solve the equation
ff
2 f
1
 x   x

1 5
x
f
f
x
 2
x
5

2
1 5
1
Note
The graph of f 1
is the reflection of the graph of f about the line y  x
(b, a)
(a, b)
Another useful fact is this: Since an invertible function must be one-to-one,
two different x values can not correspond to the same y value. Looking at
the figure, we see that this means In order for f to be invertible,
no horizontal line can intersect the graph of f more than once.
y
x
Symmetry
Even function
4
3
f  x  f  x
2
1
-2
-1
1
2
its graph is symmetric with respect to the y-axis.
3
Odd function
f  x   f  x
2
1
-2
-1
1
-1
-2
-3
its graph is symmetric about the origin.
2
Example
Determine whether each of the following functions is even,
odd neither even nor odd.
f x   x 4  3
f  x    x   3
4
f x   x 3  x
f  x    x    x 
3
 x 3
 x 3  x
 f  x
  x3  x
4
even

  f  x
odd

f  x   x 2  3x
f  x    x   3 x 
2
 x 2  3x
neither even nor odd