How many of the following functions are even [sin x is odd and cosx is even]
(a) f(x) = x2|x| (b) f(x) = ex+e−x
(c) f(x) = log[1−x1+x] (d) log(√x2+1- x)
(e) f(x) = log(x + √x2+1 (f) ax−a−x
(g) f(x) = sinx+cosx (h) sinx×(ex−e−x)
Here, we will first find f(-x). Then we will check if f(x) = f(-x) for even functions and f(x) = -f(-x) for odd functions.
(1) f(-x) = (−x)2|-x|) = x2|x|=f(x)
⇒ Even function
(2) f(-x) = e−x+e−(−x)=e−x+ex=f(x)
⇒ Even
(3) f(-x) = log (1−(−x)1−x) = log(1+x1−x
= -log(1−x1+x = -f(x)
⇒ Odd function
(4) f(-x) = log(√(−x2)+1−(−x))
= log(√(−x2)+1+ x)
= - log 1(√x2+1+x)= -log(√(x2)+1−x√(x2)+1+x√(x2)+1−x)
= -log(√(x2)+1−x√(x2)+1−x2) = -log√(x2)+1-x
f(x) = -f(x)
⇒ Odd function
(5) f(-x) = log(√(x2)+1-x)
= log√(x2)+1−x(√(x2)+1+x) × (√(x2)+1+ x)
= logx2+1−x2√(x2)+1+x
= log1(√(x2)+1+x)
= - log (√(x2)+1+ x)
(6) f(-x) = a−x - a−(−x)
= a−x - ax
= -(ax - a−x)
= -f(x)
⇒ Odd function
(7) f(-x) = sin (-x) + cos (-x)
= -sin x + cos x
=This is not equal to f(x) or f(-x)
⇒ Neither odd nor even
(8) f(-x) = sin (-x) × (e−x - e−1(−x))
= -sin x × (e−x - ex)
= sin x (ex - e−x)
= f(x)
⇒ Even function