Solveeit Logo
Login

Question

Question: Which of the following is an even function...

Which of the following is an even function

A

x(ax−1ax+1)x\left( \frac{a^{x} - 1}{a^{x} + 1} \right)

B

tan⁡x\tan x

C

ax−a−x2\frac{a^{x} - a^{- x}}{2}

D

ax+1ax−1\frac{a^{x} + 1}{a^{x} - 1}

Answer

x(ax−1ax+1)x\left( \frac{a^{x} - 1}{a^{x} + 1} \right)

Explanation

Solution

We have : f(x)=x(ax−1ax+1)f(x) = x\left( \frac{a^{x} - 1}{a^{x} + 1} \right)

f(−x)=−x(a−x−1a−x+1)=−x(1ax−11ax+1)=−x(1−ax1+ax)=x(ax−1ax+1)=f(x)f( - x) = - x\left( \frac{a^{- x} - 1}{a^{- x} + 1} \right) = - x\left( \frac{\frac{1}{a^{x}} - 1}{\frac{1}{a^{x}} + 1} \right) = - x\left( \frac{1 - a^{x}}{1 + a^{x}} \right) = x\left( \frac{a^{x} - 1}{a^{x} + 1} \right) = f(x)

So, f(x)f(x) is an even function