If-f-x-ln-a-1-1-x-b-is-an-odd-function-then-find-the-value-of-a-b-

Question Number 184594 by CrispyXYZ last updated on 09/Jan/23

If f (x) = \ln ∣ a + \frac{1}{1 - x} ∣ + b is an odd function,

then find the value of a, b .

Answered by mr W last updated on 09/Jan/23

f (x) = - f (- x)

\ln ∣ a + \frac{1}{1 - x} ∣ + b = - \ln ∣ a + \frac{1}{1 + x} ∣ - b

\ln ∣ (a + \frac{1}{1 - x}) (a + \frac{1}{1 + x}) ∣ + 2 b = 0

\ln ∣ (\frac{a + 1 - a x}{1 - x}) (\frac{a + 1 + a x}{1 + x}) ∣ + 2 b = 0

\ln ∣ (\frac{{(a + 1)}^{2} - {(a x)}^{2}}{1 - x^{2}}) ∣ + 2 b = 0

\ln ∣ a^{2} (\frac{{(1 + \frac{1}{a})}^{2} - x^{2}}{1 - x^{2}}) ∣ + 2 b = 0

s u c h t h a t t h i s h o l d s f o r a n y x,

1 + \frac{1}{a} = - 1 \Rightarrow a = - \frac{1}{2}

\ln ∣ \frac{1}{4} ∣ + 2 b = 0 \Rightarrow - 2 \ln 2 + 2 b \Rightarrow b = \ln 2

f (x) = \ln ∣ \frac{1}{1 - x} - \frac{1}{2} ∣ + \ln 2 = \ln ∣ \frac{1 + x}{1 - x} ∣ i s

o d d f u n c t i o n .

Commented by mr W last updated on 09/Jan/23

Commented by CrispyXYZ last updated on 09/Jan/23

Thanks sir! Great solution!