If the real valued function fx-ax-1xnax-1 is even and n- then least possible value of n is

∵ f(x) is even ∴ f(x)=f( x) ⇒a^x 1x^n(a^x+1)=a^ x 1( x)^n(a^ x+1) ⇒a^x 1x^n(a^x+1)= (a^x 1)( x)^n(a^x+1) ⇒1x^n= 1( x)^n ∴ least possible value for n=1 nbsp; ...

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Standard XIII

Mathematics

Even Extension of a Function

If the real v...

Question

If the real valued function $f (x) = \frac{a^{x} - 1}{x^{n} (a^{x} + 1)}$ is even and $n \in N$ , then least possible value of $n$ is

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Solution

$∵ f (x)$ is even
$∴ f (x) = f (- x)$
$\Rightarrow \frac{a^{x} - 1}{x^{n} (a^{x} + 1)} = \frac{a^{- x} - 1}{(- x)^{n} (a^{- x} + 1)}$
$\Rightarrow \frac{a^{x} - 1}{x^{n} (a^{x} + 1)} = - \frac{(a^{x} - 1)}{(- x)^{n} (a^{x} + 1)}$
$\Rightarrow \frac{1}{x^{n}} = - \frac{1}{(- x)^{n}}$
$∴$ least possible value for $n = 1$

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Even Extension of a Function

Standard XIII Mathematics

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If the real valued function f(x)=ax−1xn(ax+1) is even and n∈N, then least possible value of n is

∵f(x) is even ∴f(x)=f(−x) ⇒ax−1xn(ax+1)=a−x−1(−x)n(a−x+1) ⇒ax−1xn(ax+1)=−(ax−1)(−x)n(ax+1) ⇒1xn=−1(−x)n ∴ least possible value for n=1

If the real valued function $f (x) = \frac{a^{x} - 1}{x^{n} (a^{x} + 1)}$ is even and $n \in N$ , then least possible value of $n$ is