Question

f(x)=x-4 g(x)=x2 h(x)=3x-5 to show (fog) oh =fo(goh)​

Answer 1

Given:

f(x) = x - 4

g(x) = x²

h(x) = 3x - 5

To prove:

(fog)oh = fo(goh)

OR (fog)oh(x) = fo(goh)(x)

Proof:

Firstly,

(fog)(x) = f[g(x)]

= f(x²)

= x² - 4

Now,

LHS = (fog)oh(x)

= (fog)(3x - 5)

= (3x - 5)² - 4

= 9x² - 30x + 25 - 4

= 9x² - 30x + 21

Secondly,

(goh)(x) = g[h(x)]

= g(3x-5)

= (3x - 5)²

= 9x² - 30x + 25

Now,

RHS = fo(goh)(x)

= f(9x² - 30x + 25)

= 9x² - 30x + 25 - 4

= 9x² - 30x + 21

Since,

LHS = RHS

Hence proved.