Q. 133.5( 2 Votes )

If f, g: R

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


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