If f(x) = frac{a^x - a^{-x}}{a^x + a^{-x}},wh | vedclass

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(A) Let $y = f(x) = \frac{a^x - a^{-x}}{a^x + a^{-x}}$.Multiply the numerator and denominator by $a^x$:$y = \frac{a^{2x} - 1}{a^{2x} + 1}$.Now,solve f

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$\frac{1}{2} \log_a \left( \frac{1+x}{1-x} \right)$

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(A) Let $y = f(x) = \frac{a^x - a^{-x}}{a^x + a^{-x}}$.Multiply the numerator and denominator by $a^x$:$y = \frac{a^{2x} - 1}{a^{2x} + 1}$.Now,solve for $x$ in terms of $y$:$y(a^{2x} + 1) = a^{2x} - 1$$y \cdot a^{2x} + y = a^{2x} - 1$$1 + y = a^{2x} - y \cdot a^{2x}$$1 + y = a^{2x}(1 - y)$$a^{2x} = \frac{1+y}{1-y}$.Taking the logarithm base $a$ on both sides:$2x = \log_a \left( \frac{1+y}{1-y} \right)$$x = \frac{1}{2} \log_a \left( \frac{1+y}{1-y} \right)$.Replacing $y$ with $x$ to find the inverse function:$f^{-1}(x) = \frac{1}{2} \log_a \left( \frac{1+x}{1-x} \right)$.

If $f(x) = \frac{a^x - a^{-x}}{a^x + a^{-x}}$,where $a$ and $x$ satisfy the necessary conditions,then $f^{-1}(x) =$

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