Published by:
CGP EDU Academic Team
Published on: August 13, 2026
If g(x) is monotonically increasing and f(x) is monotonically decreasing for x ∈ R and if (gof) (x) is defined for x ∈ R, then prove that (gof)(x) will be monotonically decreasing function. Hence prove that
(g o f) (x + 1) ≤ (g o f) (x – 1).
Text Solution
Verified by ExpertsThe correct answer is:
CHECK THE SOLUTION.
g(x) is monotonically increasing
⇒ g ′ (x) ≥ 0 & f(x) is M.D. ⇒ f ′ (x) ≤ 0
(fog) (x) = 
= 
as f ′ (x) ≤ 0 & g ′ (x) ≥ 0
⇒ (fog) (x) is monotonically decreasing ⇒ f ′ (x) ≤ 0
Also x + 1 > x – 1
⇒ f(x + 1) < f(x – 1) as f(x) is M.D.
⇒ g(f(x + 1) < g(f(x – 1)) as g(x) is M. Ι .
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