the inverse hyperbolic tangent function can be expressed as $\tanh^{-1}x=\frac{1}{2}lnleft(\frac{1 + x}{1…

the inverse hyperbolic tangent function can be expressed as $\tanh^{-1}x=\frac{1}{2}lnleft(\frac{1 + x}{1 - x}\right)$, $-1<x<1$. use this formula to evaluate the following expression. round your answer to two decimal places. if the value does not exist, write dne for your answer.\n$\tanh^{-1}(0.2)$

the inverse hyperbolic tangent function can be expressed as $\tanh^{-1}x=\frac{1}{2}lnleft(\frac{1 + x}{1 - x}\right)$, $-1<x<1$. use this formula to evaluate the following expression. round your answer to two decimal places. if the value does not exist, write dne for your answer.\n$\tanh^{-1}(0.2)$

Answer

Explanation:

Step1: Substitute x value

Substitute $x = 0.2$ into the formula $\tanh^{-1}x=\frac{1}{2}\ln\left(\frac{1 + x}{1 - x}\right)$. So we get $\frac{1}{2}\ln\left(\frac{1+0.2}{1 - 0.2}\right)$.

Step2: Simplify the fraction inside the logarithm

First, simplify $\frac{1 + 0.2}{1 - 0.2}=\frac{1.2}{0.8}=\frac{3}{2}$. Then the expression becomes $\frac{1}{2}\ln\left(\frac{3}{2}\right)$.

Step3: Calculate the natural - logarithm and multiply

We know that $\ln\left(\frac{3}{2}\right)\approx0.4055$. Then $\frac{1}{2}\times0.4055 = 0.20275$. Rounding to two decimal places, we get $0.20$.

Answer:

$0.20$