How can I find the formulas for the composite functions (f º g) and (g º f) ?

Question

I need help. Can someone teach me how to do the following as a example for me. Appreciate your help.

f(x) = x - 2 ; g(x) = x^2 + 1

Answer 1

f∘g(x) = f(g(x))

=> f∘g(x) = f(x² + 1)

=> f∘g(x) = (x² + 1) − 2

=> f∘g(x) = x² − 1



g∘f(x) = g(f(x))

=> g∘f(x) = g(x − 2)

=> g∘f(x) = (x − 2)² + 1

=> g∘f(x) = x² − 4x + 4 + 1

=> g∘f(x) = x² − 4x + 5

Answer 2

there is no formula for composition of functions.

only by your understanding you have to compose

listen

f(x) = x-2, g(x) = x^2+1, the f(a)=a-2, and g(b) = b^2+1 right..!

now come to our composition

fog(x) = f{g(x)} = f{x^2+1} = (x^2+1)-2 [just like f(a) = a-2 ]

= x^2+1-2 = x^2-1

gof(x) = g{f(x)} = g{x-2} = (x-2)^2+1 [ just like g9a) = a^2+1 ]

= x^2-4x+4+1 = x^2-4x+5

Answer 3

fog(x) = f(g(x)) = f(x^2 +1) = x^2 + 1 -2 = x^2 -1

gof(x) = g(f(x)) = g(x-2) = (x-2)^2 +1 = x^2 - 4x + 4 +1 = x^2 - 4x +5