cALculus (fUnctioN)

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 #