Which one of the following is an even function?

Question

Which one of the following is an even function? (A)  f(x)=(ax+a-x/ax-a-x)  (B)  f(x)=(ax+1/ax-1)  (C)  f(x)=x(ax-1/ax+1)  (D)  f(x)= log e(x+√x2+1)

Tardigrade Admin · Accepted Answer

If f(-x)=f(x), then f(x) is an even function ∴ Putting x=-x in x(ax-1/ax+1) ∴ f(-x)=-x.(a-x-1/a-x+1) =-x.(a-x(1-ax)/a-x(1+ax)) =-x((1-ax)/(1+ax)) =x( (ax-1/ax+1) )=f(x) ∴ x(ax-1/ax+1) is an even function.

Q. Which one of the following is an even function?

$f (x) = \frac{a ^{x} + a ^{- x}}{a ^{x} - a ^{- x}}$

$f (x) = \frac{a ^{x} + 1}{a ^{x} - 1}$

$f (x) = x \frac{a ^{x} - 1}{a ^{x} + 1}$

$f (x) = lo g_{e} (x + x^{2} + 1)$

Q. Which one of the following is an even function?

f(x)=ax−a−xax+a−x​

f(x)=ax−1ax+1​

f(x)=xax+1ax−1​

f(x)=loge​(x+x2+1​)

If f(−x)=f(x), then f(x) is an even function ∴ Putting x=−x in xax+1ax−1​ ∴ f(−x)=−x.a−x+1a−x−1​ =−x.a−x(1+ax)a−x(1−ax)​ =−x(1+ax)(1−ax)​ =x(ax+1ax−1​)=f(x) ∴ xax+1ax−1​ is an even function.

$f (x) = \frac{a ^{x} + a ^{- x}}{a ^{x} - a ^{- x}}$

$f (x) = \frac{a ^{x} + 1}{a ^{x} - 1}$

$f (x) = x \frac{a ^{x} - 1}{a ^{x} + 1}$

$f (x) = lo g_{e} (x + x^{2} + 1)$