Let f and g be the functions R R given by fx 3x 5 and gx

Let f and g be the functions R R given by f(x) = 3x + 5 and g(x) = x^2 3x. Find the composite functions (a) f g (b) g f (c) g g

Given,

f(x) = 3x+5 ---(a)

g(x) = x^2 -3x ----(b)

Now,

a) f o g = f( g(x) )

Putting the function g(x) in eq (a) in place of x we get,

f (g(x)) = 3[x^2 -3x] +5 since, x=x^2 - 3x

=> 3x^2 - 9x + 5 -Ans.

b) g o f = g(f(x))

Putting the function f(x) in eq. b, in place of x we get,

g(f(x)) = [3x + 5]^2 - 3[3x +5] since, x=3x+5

=> [3x+5] [3x+5 -3] Taking [3x+5] common

=>[3x+5] [3x + 2] -Ans.

c) g o g = g(g(x))

Putting the function g(x) in its own equation in place of x, we get

g(g(x)) = [x^2 - 3x]^2 - 3[x^2 - 3x] since x=x^2 3x

=> [x^2 - 3x] [x^2 - 3x - 3] Taking x^2 - 3x common

hence, [x^2 - 3x] [x^2 - 3x - 3] -Ans