Given the function f, g and h be defined by f(x)=2x-3, g(x)=3x sqaure-4 and h=sin(x). Find the formula defining the composition function h°f°g

Question

Composition functions

Answer 1

Given the function f, g and h be defined by f(x)=2x-3, g(x)=3x sqaure-4 and h=sin(x). Find the formula defining the composition function h°f°g

f(x) = 2x - 3

g(x) = 3x^2 - 4

h(x) = sin(x)

Let k(x) = f°g    (replace all instances of x in f(x) by g(x))

Then, k(x) = (2x - 3)°(3x^2 - 4)   (replace the x in 2x by 3x^2 - 4)

k(x) = 2(3x^2 - 4) - 3

k(x) = 6x^2 - 11

Now let q(x) = h°k    (replace all instances of x in h(x) by k(x))

Then q(x) = sin(x)°(6x^2 - 11)         (replace the x in sin(x) by 6x^2 - 11)

q(x) = sin(6x^2 - 11)

And, q(x) = h°k​ = h°f°g

So, h°f°g ​= sin(6x^2 - 11)