Relation
and function
Example 1
A function g is defined by g(x)= 2x²-1 Find
(a) g(3)
g(3) = 2(3)²-1
2(9)-1= 17#
(b) g(0.5)
g(0.5)= 2(0.5)²-1
=2(0.25)-1
=0.5-1
=-0.5
(c) g(-2)
g(-2)=2(-2)²-1
=2(4)-1
=8-1
=7#
Example 2:
A function g is
defined by g(x)=x/3+1. Find
(a) g(3)
g(3)= 3/3 + 1
= 1+ 1
= 2#
Example 4:
If f(x)=x²+3x+2 x is real, express each of the
following in terms of 
:
:
(a) f(2x)
f(2x) = (2x)² + 3(2x)+ 2
= 4x²+6x+2#
(b) f(2x+1)
f(2x+1)= (2x+1)(2x+1) + 3(2x+1)+2
= 4x²+2x+2x+1+6x+3+2
= 4x²+10x
+6
= 2x²+5x+3#
Composite
function
EXAMPLE 1
Given that f(x)=3x + 1 and g(x)= x/2; Find the value of
composite function of gf(x), ff(x) and fg(2)
SOLUTION
Study the given basic
function —-> since both are basic functions,
then this require solution under TYPE 1
gf(x) =
g(3x + 1) by virtue of
replacing 3x + 1 into f(x)
= (3x + 1)/2 #
ff(x) is sometimes written as f²(x)
= f(3x + 1)
= 3(3x + 1) + 1
= 9x + 4 #
Before we solve the composite function of fg(2), first we need
to solve g(2) as a function
g(2)
= 2/2
= 1
Replace the newly found value of g(2)=1 into fg(2) and that will
give a new function of f(1)
fg(2)
= f(1)
= 3(1) + 1
= 4 #
EXAMPLE 2
Given that the function f(x) = x + 4 and composite function of
fg(x) = 5 – 3x, then find the value of function g(3)
SOLUTION
Study the given basic
function —-> since only 1 basic function is
given and the other unknown basic function mentioned 1st in the given composite
function, then this require solution under TYPE 2
To find the value of g(3), we must first find the function of
g(x)
fg(x)
= g(x) + 4 –> from
basic function of f(x)
fg(x)
= 5 – 3x —-> from the
given composite function
g(x) + 4
= 5 – 3x
g(x)
= 5 – 3x – 4
= 1 – 3x #
JUSTIFY YOUR ANSWER
fg(x)
= f(1 –
3x) = (1 – 3x) +
4
= 5 – 3x
From the findings above where g(x) = 1 – 3x
Therefore g(3)
= 1 – 3(3)
= - 8 #
EXAMPLE 3
Given that the function f(x) = 2x and composite function of
gf(x) = 3/x, then find the value of function g(3)
SOLUTION
Study the given basic
function —-> since only 1 basic function is
given and the other unknown basic function is mentioned 2nd in the given
composite function, then this require solution under TYPE 3
To find the function of g(x), we must first redo the f(x) in
order to find the value of x. This need to be done as gf(x) = 3/x indicates the
value of x in the function g(x) has been utilized in the composite function.
Let f(x)
= y; now y = 2x
x
= y/2
Now we can replace the
value of x in the composite function of gf(x) that is now known as g(y)
gf(x)
= g(y)
= 3/y/2
= 3 x 2/y
= 6/y
Therefore
g(x)
= 6/x #
JUSTIFY YOUR ANSWER
gf(x)
= g(2x)
= 6/2x
= 3/x
From the findings above where g(x) = 6/x
Therefore g(3) = 6/3
= 2 #