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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).
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Solution
Given: f(x) = |x| + x
and g(x) = |x| -x, ∀x ∈ R
fog = f(g(x)) = | g (x) | + g(x)
= ||x| − x|+(|x| − x)
Therefore,
f( g(x)) = `{ (0 x ≥ 0), (4x x <0):}`
f( g(x)) = `{ (4x x > 0), (0 x ≥ 0):}`
gof = g (f(x)) = |f(x)| − f (x)
= ||x|+x| − (|x|+x)
g(f(x)) = `{(0 x ≥ 0), (0 x < 0):}`
Therefore, g (f(x)) = gof = 0
Now, fog(−3) =(4)(−3) = −12 (since, fog = 4x for x < 0)
fog (5) = 0 (since, fog = 0 for x ≥ 0)
gof(−2) = 0 (since, gof = 0 for x < 0)
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