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

Question

Philoid · Accepted Answer

Domain of f(x) and g(x) is R.

Range of f(x) = [0, ∞) and range of g(x) = [0, ∞)

As, range of f ⊂ Domain of g and range of g ⊂ Domain of f

So, gof and fog exists

Now,

fog(x) = f(g(x)) = f(|x|–x)

⇒ fog(x) = ||x|–x| + |x|–x

As, range of g(x) ≥ 0 so, ||x|–x| = |x|–x

So, fog(x) = ||x|–x| + |x|–x = |x|–x + |x|–x

⇒ fog(x) = 2(|x|–x)

Also,

gof(x) = g(f(x)) = g(|x| + x) = ||x| + x| – (|x| + x)

As, range of f(x) ≥ 0 so, ||x| + x| = |x| + x

So, gof(x) = ||x| + x| – (|x| + x) = |x| + x – (|x| + x) = 0

Thus, gof(x) = 0

Now, fog(– 3) = 2(| – 3|–(– 3)) = 2(3 + 3) = 6,

fog(5) = 2(|5| – 5) = 0, gof(– 2) =0

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

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

fog (5) and gof(–2).