For the pair of functions fx x 3 and gx x2 x 6 find the

For the pair of functions, f(x) = x + 3 and g(x) = x^2 + x - 6, find the following. (f g)(x) (f g)(2) (g f)(x) (g f)(2) (f g)(x) (Simplify your answer.)

Solution

Given f(x) = x+3

g(x) = x²+x-6

a)(fog)(x):

(fog)(x) = f(g(x))

= f(x²+x-6)

f(x) = x+3

f(x²+x-6) = x²+x-6 +3

= x²+x-3

Therefore, (fog)(x)=x²+x-3

b) (fog)(2):

(fog)(x)=x²+x-3

(fog)(2)= 2²+2-3

= 4+2-3

= 3

Therefore, (fog)(2) = 3

c)(gof)(x):

(gof)(x) = g(f(x))

= g(x+3)

g(x) = x²+x-6

g(x+3) = (x+3)²+(x+3)-6

= x²+3²+6x+x+3-6

= x²+7x+6

Therefore, (gof)(x) = x²+7x+6

d) (gof)(2):

(gof)(x) = x²+7x+6

(gof)(2) = 2²+7(2)+6

= 4+14+6

= 24

Therefore, (gof)(2) = 24