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Given f(x) = ((1+x)/(1-x)).

Mathematics

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Graph of f(x) = (1 + x)/(1 − x)

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Key points of f(x) = (1 + x)/(1 − x)
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x-intercept−10
y-intercept01

Step 1: Show that f(x)f(x) is an odd function.
A function f(x)f(x) is odd if f(−x)=−f(x)f(-x) = -f(x).
Given f(x)=ln⁡(1+x1−x)f(x) = \ln\left(\frac{1+x}{1-x}\right).
Substitute −x-x into the function:

f(−x)=ln⁡(1+(−x)1−(−x))=ln⁡(1−x1+x)f(-x) = \ln\left(\frac{1+(-x)}{1-(-x)}\right) = \ln\left(\frac{1-x}{1+x}\right)

Using the logarithm property ln⁡(ab)=−ln⁡(ba)\ln\left(\frac{a}{b}\right) = -\ln\left(\frac{b}{a}\right):

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