Question

A function f is said to be invertible with respect to integration over the interval (a,b] if and only if f is one-to-one and continuous on the interval (a,0), and in addition (2) de f(x) dx. In the list below, some functions are described either by their rules or by their graphs. Select all the functions which are invertible with respect to integration over the interval (0,1). (A) f(x) = 1 + cos(-AI) (D) S(r) = 1 + cos(-22) (B) y (0.1) 1 (1.1) (0.1) (1,1) 2 (0,0) (1.0) (0,0) (1.0) (C) (F) (0.1) (1,1) (0,1)] 2 (0,0) (1.0) (0,0) (1,0)

Answer 1

x + (A) f(x) = x + cos GTK) Cos (TTC) cos (-o) = coso. as so f(x) x + cos (+12) one - one check if f is lete now let x + cos TTx = x₂ + ws Tx2 x x COSTTR, COST7, here H in [al Costix is decreasing function & increasing function th x is so if x 2270 x, > x2 > & Los Tx₂ - Costa x, & x2 f(n) = x + hence can find & XL such that & f(x) = f(x2) cos (-T7x) is not hence it doesn't satisfy defination of invertible w.rit integration . One - one

(B) clearly here (0, 1) is one - one А ze 1001 (1,0) we f from graph as for fle) = f(ou) get ni = RL area shaded are a of square = B. zI-B oy consider B. & area so S't du a od pidza interpreting integral as 1x1 - B=1-B under area the curve, hence for f it's X - are a curve & f its area curve between axis while for between In y-axis. also f is continion hence it statifies defination. hence w.rot. integration . invertible

(o the as is not one - one let given graph of the function for get f(x) = f (x2) 1 hence satisfy the defination. say 2 x, it doesn't x + f(x) cos (-Tac 2) cos (T1x²) as Cos (-o) = coso it is similar to part (A) henne it is one - one hence it doesn't satisfy defination. f(x) is not invertible under integration The hence function argument is similar to part (B) it is invertible und wirt. integration (F) we one one, 1 Here also give very same argument as part (3) it is Continious A s'tane off dx , do it is invertible integration - with respect to

So the functions which under are invertible integration (B) (E) & (f)
.

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