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Question
Let f : R → R and g : R → R be defined by f(x) = x + 1 and g (x) = x − 1. Show that fog = gof = IR.
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Solution
Given, f : R → R and g : R → R
⇒ fog : R → R and gof : R → R (Also, we know that IR : R → R)
So, the domains of all fog, gof and IR are the same.
(fog) (x) = f (g (x)) = f (x−1) = x−1+1= x = IR (x) ... (1)
(gof) (x) = g (f (x)) = g (x+1)= x+1−1 = x=IR (x) ... (2)
From (1) and (2),
(fog) (x) = (gof) (x) = IR (x), ∀x ∈ R
Hence, fog = gof = IR
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