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Congruency

31. if f(x) i...

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31. if f(x) is given by

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Q. If f : R → R is given by f(x) = 2x + |x|, then f(2x) + f(–x) + 4x = _______________.

f (x) = ∣ ∣ ∣ ∣ \begin{matrix} a & - 1 & 0 a x & a & - 1 a x^{2} & a x & a \end{matrix} ∣ ∣ ∣ ∣

f (2 x) - f (x)

is divisible by

Q. A function f:

R \to R

sin x cos y [f (2 x + 2 y) - f (2 x - 2 y)] = cos x sin y [f (2 x + 2 y) + f (2 x - 2 y)]

f^{'} (0) = \frac{1}{2}

f_{1} (x)

f_{2} (x)

are defined on domains

D_{1}

D_{2}

respectively, then

f_{1} (x) + f_{2} (x)

D_{1} \cap D_{2}

.

f (x) = ∣ ∣ ∣ ∣ \begin{matrix} a & - 1 & 0 a x & a & - 1 a x^{2} & a x & a \end{matrix} ∣ ∣ ∣ ∣

, by using properties of determinants, find the value

f (2 x) - f (x)

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