What is the inverse of the function?
For the following two (02) items: Consider the function f(x)=\frac{10^{x}-10^{-x}}{10^{x}+10^{-x}}.
- A. \log_{10}(2x-1)
- B. \frac{1}{2}\log_{10}(2x-1)
- C. \frac{1}{4}\log_{10}\left(\frac{2x}{2-x}\right)
- D. \frac{1}{2}\log_{10}\left(\frac{1+x}{1-x}\right) ✓
Correct Answer: D. \frac{1}{2}\log_{10}\left(\frac{1+x}{1-x}\right)
Explanation
Let y = \frac{10^x - 10^{-x}}{10^x + 10^{-x}} = \frac{10^{2x}-1}{10^{2x}+1}. Rearranging to solve for 10^{2x} yields 10^{2x}(1-y) = 1+y, so 10^{2x} = \frac{1+y}{1-y}. Taking the base-10 logarithm on both sides gives 2x = \log_{10}\left(\frac{1+y}{1-y}\right). Swapping variables gives f^{-1}(x) = \frac{1}{2}\log_{10}\left(\frac{1+x}{1-x}\right).
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