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Question

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+ex

(c) f(x) = log[1x1+x] (d) log(x2+1- x)

(e) f(x) = log(x + x2+1 (f) axax

(g) f(x) = sinx+cosx (h) sinx×(exex)


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Solution

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) = ex+e(x)=ex+ex=f(x)

Even

(3) f(-x) = log (1(x)1x) = log(1+x1x

= -log(1x1+x = -f(x)

Odd function

(4) f(-x) = log((x2)+1(x))

= log((x2)+1+ x)

= - log 1(x2+1+x)= -log((x2)+1x(x2)+1+x(x2)+1x)

= -log((x2)+1x(x2)+1x2) = -log(x2)+1-x

f(x) = -f(x)

Odd function

(5) f(-x) = log((x2)+1-x)

= log(x2)+1x((x2)+1+x) × ((x2)+1+ x)

= logx2+1x2(x2)+1+x

= log1((x2)+1+x)

= - log ((x2)+1+ x)

(6) f(-x) = ax - a(x)

= ax - ax

= -(ax - ax)

= -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) × (ex - e1(x))

= -sin x × (ex - ex)

= sin x (ex - ex)

= f(x)

Even function


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