

Introduction
to Functions 
Function definition, notation and
terminology 
Domain, range and
codomain 
Evaluating
a function 
Composition
of functions (a function of a function) 
Inverse
function






Function definition, notation and
terminology 
A function
f
is
a relation between two sets, called the domain
and the range,
such that to each element x of the domain, there is assigned
exactly one element f
(x) of the range. 
We also say that
a
function is an expression or a rule that associates
each element of the domain with a unique element
of the codomain. 
In
the function notation y = f
(x),
x
is the independent variable or argument and y
is the dependent variable or a function of the variable x,
where f
is a rule of association. In the notation y = f
(x)
we call y
the value of f
at x. 
Thus, ƒ
represents all the operations which should be performed to
evaluate the function at a particular value. 

Domain, range and
codomain 
The domain is
the set of values of the independent variable of a given function,
i.e., the set of all first members of the ordered pairs (x,
f (x))
that
constitute the function. 
The range is
the set of values that given function takes as its argument varies
through its domain. It is the image of the domain. 
The codomain
is the set within which the values of a function lie, as opposed to
the range, which is the set of values that the function actually
takes. 
Therefore, the range must be a subset of,
but may or may not be identical with the codomain. 
We will only consider realvalued
functions of a real variable. 

Evaluating
a function 
Evaluating
a function means finding f
(x)
at some
specific value of x.
So, evaluating a function at a constant or a variable involves substituting the constant
or the variable into the expression of the function and
calculate its value. 

Example: Given
f (x)
= x^{2} +
4x  1
find, a) f
(1) and
b) f (x
+ 2). 
Solution:
a)
f (1)
= (1)^{2} +
4(1)
 1
= 1

4  1
= 6 
b) f (x
+ 2)
= (x
+ 2)^{2} +
4(x + 2)  1
= (x^{2}
+
4x +
4) +
4x + 8  1
= x^{2}
+ 3 

Composition
of functions (a function of a function) 
Evaluation
of a function at the value of another (or the same) function is
called the composition of functions, denoted as (ƒ
o
g)(x) = ƒ(g(x)). 
Thus,
the composition is the operation that forms a single function
from two given functions by plugging the second function into
the first for any argument. 
The composition
of functions is only defined if the range of the first is
contained in the domain of the second function. 

Examples: Given
f (x)
= 
x^{2} +
4x  1
and g (x)
= 
x
+ 1
find; 
a) f (g
(x)),
b) g (f
(x)),
c) g (g
(x))
and d) f
(g (1)). 

Solutions:
a) f
(g (x))
= ƒ(
x
+ 1)
= 
(
x
+ 1)^{2} +
4(x
+ 1)
 1
= 
x^{2}

2x + 2 
b) g (f
(x))
= g (
x^{2} +
4x  1)
= 
(x^{2} +
4x  1)
+ 1
= x^{2} 
4x + 2 
c) g (g
(x))
= g (
x
+ 1)
= 
(
x
+ 1)
+ 1
= x 
d) f (g
(1))
= f ( 
(1)
+ 1)
= f (2)
= 
2^{2}
+ 4 · 2  1
= 3 

Inverse
function

The inverse function, usually written
f ^{1}, is the function whose
domain and the range are respectively the range and domain of a given function
f, that is 
f
^{1}(x)
= y if and only if
f (y)
= x . 
Thus, the
composition of the inverse function and the given function returns
x, which is called the
identity function, i.e., 
f
^{1}(f
(x))
= x and
f (f
^{1}(x))
= x. 
The inverse of a function undoes the procedure
(or function) of the given function. 
A pair of inverse functions is in
inverse relation. 
Example: If
given
f (x)
= log_{2 }x
then f ^{1}(x)
= 2^{x}
since, 


Therefore,
to obtain the inverse of a function y =
f (x),
exchange the variables x
and y,
i.e., write x = f
(y)
and solve for y.
Or form the composition f
(f
^{1}(x))
= x and solve
for f ^{1}. 

Example: Given
y = f
(x)
= log_{2 }x determine
f ^{1}(x). 
Solution:
a) Rewrite
y = f
(x)
= log_{2 }x
to x =
log_{2 }y
and solve for y,
which gives y =
f ^{1}(x)
= 2^{x}. 
b) Form f (f
^{1}(x))
= x that
is, log_{2
}(f ^{1}(x))
= x and
solve for f ^{1}, which
gives f ^{1}(x)
= 2^{x}. 

The
graphs of a pair of inverse functions are symmetrical with
respect to the line
y
= x. 









Intermediate
algebra contents 



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