Differential Calculus 201-NYA-05
Vincent Carrier
Real-Valued Functions
A function from a set D to a set R, f : D → R, is a rule that assigns to each member
of D one and only one member of R.
D
R
D
1 rQ
r
1Q
2
2 r
3
r
3 4
Q
Q -r
r
Q
5
QQ
QQ
r
sr
R
6
3
r4
Q
Q
3
Q
1r 5
Q
Q
Q
r
sr
Q
6
function
not a function
The set D is called the domain of f and the set R the range of f .
A real-valued function is a function f : D → R where D and R are subsets of R.
A real-valued function is defined using an algebraic expression:
f (x) = x2 − 3x + 2,
f (x) = 3x − 2,
f (x) =
It can be described by a graph in the xy-plane where y = f (x). Thus,
graph of f = {(x, f (x)) : x ∈ D}.
Here is an example where D = [a, b] and R = [c, d].
y 6
d
c
-
a
b
x
1
.
x
Not every curve in the plane represents the graph of a real-valued function.
Vertical Line Test: A curve in the xy-plane represents the graph of a real-valued
function if no vertical line intersects the curve more than once.
y 6
y 6
-
-
x
x
function
not a function
Consider a function f : D → R. We say that f is a one-to-one function if
x1 6= x2
f (x1 ) 6= f (x2 )
implies
for all x1 , x2 ∈ D.
There is a simple way to check if a function is one-to-one.
Horizontal Line Test: A curve in the xy-plane represents the graph of a one-to-one
function if no horizontal line intersects the curve more than once.
y 6
y 6
-
-
x
x
one-to-one function
not a one-to-one function
A one-to-one function f has an inverse function f −1 : R → D such that
y = f (x)
if and only if
x = f −1 (y)
This is the topic of another document.
for all x ∈ D, y ∈ R.