Composite Functions - Class 10 Solutions

Overview: If f:A->B and g:B->C are two defined functions then the composite function of f and g is the function gof or gf that is defined from A to C.


Q1. If f(x) = 2x + 2 and fog(x) = 8x - 13 then find the value of x such that gof(x) = 20.
Solution:
Given,

f(x) = 2x + 2
fog(x) = 8x + 13
gof(x) = 20

To find: x = ?

We know,
fog(x) = 8x + 13
or, f{g(x)} = 8x + 13
or, 2{g(x)} + 2 = 8x + 13
or, 2{g(x)} + 2 - 2 = 8x + 13 - 2
or, 2{g(x)} = 8x + 11
[Dividing whole equation by 2]
or, 2{g(x)} / 2 = (8x + 11)/2
So, g(x) = (8x + 11)/2

Also,
gof(x) = 20
or, g{f(x)} = 20
or, g{2x + 2} = 20
or, {8(2x +2) + 11}/2 = 20
or, 8(2x + 2) + 11 = 20×2
or, 16x + 16 + 11 = 40
or, 16x = 40 - 16 - 11
or, 16x = 40 - 27
or, 16x = 13
So, x = 13/16

Therefore, the required value of x when gof(x) = 20 is 13/16.



 Q2. If g(x) = 2x and fog(x) = 6x - 2 then find the value of x such that gof(x) = 10.
Solution:
Given,

g(x) = 2x
fog(x) = 6x - 2
gof(x) = 10
To find: x = ?

We know,
fog(x) = f{g(x)}
or, 6x - 2 = f{2x}
[Dividing x in each side by 2]
or, 6x/2 - 2 = f{2x/2}
or, 3x -2 = f(x)
So, f(x) = 3x - 2

Also,
gof(x) = g{f(x)}
or, g{f(x)} = 10
or, g{3x - 2} = 10
or, 2(3x - 2) = 10
or, 3x - 2 = 10/2
or, 3x - 2 = 5
or, 3x = 5 + 2
or, 3x = 7
So, x = 7/3

Therefore, the required value of x such that gof(x) = 10 is 7/3.


Q3. Find fog and gof when f(x) = 4x + 5 and g(x) = 8x - 7

Solution:
Given,

f(x) = 4x + 5
g(x) = 8x - 7

To find: fog(x) = ? and gof(x) = ?

We know,
fog(x) = f{g(x)}
= f{8x - 7}
= 4(8x - 7) + 5
= 32x - 28 + 5
= 32x - 23

Also,
gof(x) = g{f(x)}
= g{4x + 5}
= 8(4x + 5) - 7
= 32x + 40 - 7
= 32x + 33

Therefore, the required fog(x) and gof(x) of the given functions f(x) and g(x) are 32x - 23 and 32x + 33, respectively.


#SciPiPupil