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ADDITIONAL MATHEMATICS : FUNCTION

SOLVING COMPOSITE FUNCTION

Basically, this composite could come in various formats as the term composite indicates a combination of 2 or more of various functions. However, all these composite functions could be categorized under 3 different types which are fondly known as TYPE 1, TYPE 2 & TYPE 3. These functions are catogarised into 3 different types as it requires different steps to solve every question. To differentiate between one to another, STUDY THE BASIC FUNCTIONS GIVEN. The following are the 3 basic examples that represent the possible question under these circumstances.

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 #

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