34. A function F(x) satisfies the functional equation x2f(x) + f(1-x) =2x-x4 for all real x. F(x) must be 1. x2 2. 1-x2 3. 1+x2 4. x2+x+

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Q. Given that

x^{2} f (x) + f (1 - x) = 2 x - x^{4}

Prove that

f (x) = 1 - x^{2}

If f is a real function satisfying $f (x + \frac{1}{x}) = x^{2} + \frac{1}{x^{2}}$ for all $x ϵ R - {0}$ , then write the expression for f(x).

Q. A function f satisfies the condition

f (x + \frac{1}{x}) = x^{2} + \frac{1}{x^{2}}, x \neq 0.

Determine f(2)

Q. The function

f (x) = {\begin{matrix} x^{4} + x^{2} - x + 2 & f o r & x \leq 1 3 x^{3} - x^{2} + x & f o r & x > 1 \end{matrix}

Find the domain of each of the following real valued functions of real variable :
(i) $f (x) = \frac{1}{x}$
(ii) $f (x) = \frac{1}{x - 7}$
(iii) $f (x) = \frac{3 x - 2}{x + 1}$
(iv) $f (x) = \frac{2 x + 1}{x^{2} - 9}$
(v) $f (x) = \frac{x^{2} + 2 x + 1}{x^{2} - 8 x + 12}$

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