1.

Let f : R → R : f(x) = (2x + 1) and g : R → R : g(x) = (x2 - 2). Write down the formulae for (i) (g o f) (ii) (f o g) (iii) (f o f) (iv) (g o g)

Answer»

(i) g o f

To find: g o f

Formula used: g o f = g(f(x))

Given: (i) f : R → R : f(x) = (2x + 1)

(ii) g : R → R : g(x) = (x2 - 2)

Solution: We have,

g o f = g(f(x)) = g(2x + 1) = [ (2x + 1)2 – 2 ]

⇒ 4x2 + 4x + 1 – 2

⇒ 4x2 + 4x – 1

g o f (x) = 4x2 + 4x – 1

(ii) f o g

To find: f o g

Formula used: f o g = f(g(x))

Given: (i) f : R → R : f(x) = (2x + 1)

(ii) g : R → R : g(x) = (x2 - 2)

Solution: We have,

f o g = f(g(x)) = f(x2 - 2) = [ 2(x2 - 2) + 1 ]

⇒ 2x2 - 4 + 1

⇒ 2x2 – 3

f o g (x) = 2x2 – 3

(iii) f o f

To find: f o f

Formula used: f o f = f(f(x))

Given: (i) f : R → R : f(x) = (2x + 1)

Solution: We have,

f o f = f(f(x)) = f(2x + 1) = [ 2(2x + 1) + 1 ]

⇒ 4x + 2 + 1

⇒ 4x + 3

f o f (x) = 4x+ 3

(iv) g o g

To find: g o g

Formula used: g o g = g(g(x))

Given: (i) g : R → R : g(x) = (x2 - 2)

Solution: We have,

g o g = g(g(x)) = g(x2 - 2) = [ (x2 - 2)2 – 2]

⇒ x4 -4x2 + 4 - 2

⇒ x 4 -4x2 + 2

g o g (x) = x4 -4x2 + 2


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