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Determine the inverse of each function: (i) f(x) = ln(x + sqrt(x² + 1)); (ii) f(x) = 2^(x/(x-1)); (iii) y = (10^x - 10^-x)/(10^x + 10^-x).

1. (i) (e^x - e^-x)/2; (ii) log2(x)/(log2(x) - 1); (iii) (1/2)*log10((1+x)/(1-x))
2. (i) (e^x + e^-x)/2; (ii) x/(x-1); (iii) log10((1+x)/(1-x))
3. (i) sinh(x); (ii) log2(x/(x-1)); (iii) (1/2)*ln((1+x)/(1-x))
4. (i) e^x - 1; (ii) 1/(1 - log2 x); (iii) tanh(x)

Correct answer: (i) (e^x - e^-x)/2; (ii) log2(x)/(log2(x) - 1); (iii) (1/2)*log10((1+x)/(1-x))

Solution

(i) y = ln(x + sqrt(x²+1)) is arcsinh(x); its inverse is sinh(y) = (e^y - e^-y)/2. (ii) y = 2^(x/(x-1)) -> log2 y = x/(x-1) -> solve: x = log2(y)/(log2(y) - 1). (iii) y = (10^x - 10^-x)/(10^x + 10^-x) = tanh in base 10; let u = 10^(2x): y = (u-1)/(u+1) -> u = (1+y)/(1-y) -> 2x = log10((1+y)/(1-y)) -> x = (1/2)*log10((1+y)/(1-y)).

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