COMPOSITION OF FUNCTIONS

About the topic "Composition of functions"

Composition of functions :

It is an operation being used to combine the given two functions.

Let f(x) and g(x) be the two functions.

The formula for composition of functions is given below in different forms.

f∘g  =  f[g(x)]

f∘g(x)  =  f[g(x)]

f∘g(x²)  =  f[g(x²)]

Composition of functions - Practice problems

Problem 1 :

If f(x)  =  2x² + 3 and g(x)  =  x + 2, find  fg.

Solution :

fg  =  f[g(x)]

fg  =  f[x + 2]

fg  =  2(x + 2)² + 3

fg  =  2(x² + 2.x.2 + 2²) + 3

fg  =  2(x² + 4x + 4) + 3

fg  =  2x² + 8x + 8 + 3

fg  =  2x² + 8x + 11

Problem 2 :

If f(x)  =  5x and g(x)  =  x+2, find  fg (x²).

Solution :

fg (x²)  =  f[g(x²)]

fg (x²)  =  f[x² + 2]

fg(x²)  =  5(x² + 2)

fg(x²)  =  5x² + 10

Problem 3 :

If f(x)  =  5x + 3 and g(x)  =  7x - 2, find  fg(3).

Solution :

fg(3)  =  f[g(3)]

fg(3)  =  f[7(3) + 2]

fg(3)  =  f(23)

fg(3)  =  5(23) + 3

fg(3)  =  115 + 3

fg(3)  =  118

Problem 4 :

If f(x)  =  x -5 and g(x)  =  2x + 3, verify  fg  =  gf

Solution :

fg  =  f[g(x)]

fg  =  f[2x + 3]

fg  =  (2x + 3) - 5

fg  =  2x + 3 - 5

fg  =  2x - 2 ------->(1)

gf  =  g[f(x)]

gf  =  g[x - 5]

gf  =  2(x-5) + 3

gf  =  2x - 10 + 3

gf  =  2x - 7 ------->(2)

From (1) and (2), it is clear that  f≠  gf

Problem 5 :

Let f(x)  =  x + k and g(x)  =  7x. If  fg(2) =  7, find the value of "k".

Solution :

fg(2)  =  7

f[g(2)]  =  7

f[7(2)]  =  7

f(14)  =  7

14 + k  =  7

k  =  -7

After having gone through the stuff given above, we hope that the students would have understood "Function composition".

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