3. Consider f: I R. Let J be an interval containing f(I). Let g : J → R. Consider the composition of f and g, i.e. gof:I R. Show that: (1) If both f and g are strictly monotone increasing, then gof : I R is strictly monotone increasing. (2) If both f and g are strictly monotone decreasing, then gof: I -→ R is strictly monotone increasing. (3) If one of f and g is strictly monotone increasing and the other is strictly monotone decreasing, then gof: I→R is strictly monotone decreasing.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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3. Consider f: I R. LetJ be an interval containing f(I). Let g:
Consider the composition of f and g, i.e. gof: I-→ R. Show that:
(1) If both f and g are strictly monotone increasing, then g of: I→R is
strictly monotone increasing.
(2) If both f and g are strictly monotone decreasing, then gof: I→R is
strictly monotone increasing.
(3) If one of f and g is strictly monotone increasing and the other is strictly
monotone decreasing, then gof:I →R is strictly monotone decreasing.
J R.
Transcribed Image Text:3. Consider f: I R. LetJ be an interval containing f(I). Let g: Consider the composition of f and g, i.e. gof: I-→ R. Show that: (1) If both f and g are strictly monotone increasing, then g of: I→R is strictly monotone increasing. (2) If both f and g are strictly monotone decreasing, then gof: I→R is strictly monotone increasing. (3) If one of f and g is strictly monotone increasing and the other is strictly monotone decreasing, then gof:I →R is strictly monotone decreasing. J R.
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