FINDING THE MISSING VALUE IN COMPOSITION OF TWO FUNCTIONS

Question 1 :

Find a number b such that f ◦ g = g ◦ f , where

f(x) = 2x + b and g(x) = 3x + 4.

Solution :

L.H.S 

f ◦ g (x)  =  f [g(x)]

  =  f(3x + 4)

  =  2(3x + 4) + b

  =  6x + 8 + b  ------(1)

R.H.S :

  g ◦ f (x)  =  g [f(x)]

  =  g(2x + b)

  =  3(2x + b) + 4

  =  6x + 3b + 4  ------(2)

(1)  =  (2) 

6x + 8 + b  =  6x + 3b + 4

b - 3b  =  4 - 8

-2b  =  -4

b  =  2

So, the value of b is 2.

Question 2 :

Find a number c such that f ◦ g = g ◦ f , where

f(x) = 5x − 2 and g(x) = cx − 3.

Solution :

L.H.S 

f ◦ g (x)  =  f [g(x)]

  =  f(cx - 3)

  =  5(cx - 3) - 2 

  =  5cx - 15 - 2

=  5cx - 17  ------(1)

R.H.S :

  g ◦ f (x)  =  g [f(x)]

  =  g(5x - 2)

  =  c(5x - 2) - 3

  =  5cx - 2c - 3  ------(2)

(1)  =  (2) 

5cx - 17  =  5cx - 2c - 3

2c  =  -3 + 17

2c  =  14

c  =  14/2  =  7

c  =  7

So, the value of c is 7.

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