First slide

Functions (XII)

Question

If $2 f (x^{2}) + 3 f (\frac{1}{x^{2}}) = x^{2} - 1$ , then $f (x^{2})$ is

Moderate

A

$\frac{(1 - x^{4})}{5 x^{2}}$
B

$\frac{(1 - x^{2})}{5 x}$
C

$\frac{5 x^{2}}{(1 - x^{4})}$
D

none of these

Solution

$\begin{array}{l} 2 f (x^{2}) + 3 f (1 / x^{2}) = x^{2} - 1 \\ \Rightarrow 2 f (1 / x^{2}) + 3 f (x^{2}) = 1 / x^{2} - 1 \end{array}$
Subtracting, $f (x^{2}) - f (1 / x^{2}) = 1 / x^{2} - x^{2}$
$\Rightarrow f (x^{2}) = 1 / x^{2} .$

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