Department of Mathematics, K.T.H.M. College, Nashik

mathsbeauty

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Registered: 11-06-2015

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Function on space of differntiable functions (Part B: NET June-2015)

(Part B Question: NET June-2015) Let \(V \) be the space of twice differentiable functions on \(\mathbb{R} \) satisfying \( f''-2f'+f =0\). Define \( T: V \to \mathbb{R}^2 \) by \(T(f)=(f'(0),f(0)) \). Then T is
A. one-one and onto
B. one-one but not onto
C. onto but not one-one
D. neither one--one nor onto

\(\textbf{Option C}\)
The general solution of differential equation is \(f(x) = c_1e^x+c_2xe^x \).
So the function \(T \) is
\(T(f) = (f'(0),f(0)) = (c_1+c_2,c_1) \)
This function is clearly onto but not one-one.