Answer to Question #97128 in Calculus for Anand

Question

Check whether the given function is odd or even
f(x) = ln|(1-e⁻ˣ)/(1+e⁻ˣ)|

please its urgent answer it.

Answer 1

We are going to check the given function is Odd (or) Even.

A function $f(X)$ is said to be Odd function if

$${f(-X)} = - f(X)$$

A function $f(X)$ is said to be Even function if

$${f(-X)} = + f(X)$$

The given function is

$$f(X) = ln | \frac { 1 - e^{-X}} { 1 + e^{-X}}|$$

Now plug X = - X in the function $f(X)$:

$$f(- X) = ln |\frac {1- e^ {-(-X)}} {1 + e^ {-(-X)}}|$$

$$f(- X) = ln |\frac {1- e^ {(+X)}} {1 + e^ {(+X)}}|$$

$$f(- X) = ln |\frac {1- e^ {(X)}} {1 + e^ {(X)}}|$$

Now, divide the numerator and denominator by e^X

$$f(- X) = ln |\frac {\frac {1} { e^ {X}} -1} {\frac {1} { e^ {X}} +1}|$$

$$f(- X) = ln |\frac { e^ {-X} -1} { { e^ {-X}} +1}|$$

$$f(- X) = ln |\frac { -(-e^{-X} +1)} { { e^ {-X}} +1}|$$

$$f(- X) = ln |\frac { (-e^ {-X} +1)} { { e^ {-X}} +1}|$$

Since $| - a | = |a|$,

$$f(- X) = ln |\frac { (1 -e^ {-X})} { { 1 + e^ {-X}} }| = f(X)$$

Thus, it is an Even function.

Answer: The given function is Even