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Given f(x) = x2 + 4; g(x) = 3x – 2; h(x) = x – 5. Show that the composition of functions is associative.

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f(x) = x2 + 4 ; g(x) – 3x – 2; h(x) = x – 5

To prove fo (goh) = (fog) oh

L.H.S. fo (goh)

goh = g[h(x)]

= g(x – 5)

= 3(x – 5) – 2

= 3x – 15 – 2

goh = 3x – 17

fo (goh) = f [goh (x)]

= f(3x – 17)

= (3x – 17)2 + 4

= 9x2 + 289 – 102 x + 4

= 9x2 – 102x + 293 … (1)

R.H.S. = (fog) oh

fog – f[g(x)]

= f(3x - 2)

= (3x – 2)2 + 4

= 9x2 + 4 – 12x + 4

= 9x2 – 12x + 8

(fog) oh = fog [h(x)]

= fog (x – 5)

= 9(x – 5)2 – 12 (x – 5) + 8

= 9(x2 + 25 – 10x) – 12x + 60 + 8

= 9x2 + 225 – 90x – 12x + 60 + 8

= 9x2 – 102x + 293 ….(2)

From (1) and (2) we get fo (goh) = (fog) oh.

Composition of function is associative

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