Question

If f, g : R → R be two functions defined as f(x) = |x| + x and g(x) = |x| – x, ∀ x ∈ R. Then find fog and gof. Hence find fog (–3), fog(5) and gof (–2).

Answer 1

Here,

f(x) = |x| + x can be written as

And g(x) = |x| - x, can be written as

Therefore, gof is defined as

For x ≥ 0, gof(x) = g(f(x))

⇒ gof (x) = g(2x) = 0

and for x ≥ 0,gof(x) = g(f(x)) = g(0) =0

Hence, gof (x) = 0 ∀ x ∈ R.

Again,

fog is defined as

For x ≥ 0, fog(x) = f(g(x)) = f(0) = 0

and for x < 0, 

fog(x) = f(g(x)) = f(- 2x)

= 2(- 2x) = - 4x

Hence,

2nd part

fog(5) = 0 [∵ 5 ≥ 0]

fog(- 3) = - 4 x (- 3) = 12 [∵ 3 < 0]

gof(-2) = 0