Which of the following is an even funciton Here [.] denotes the greatest inetger function and fis any function.
[x]−x
f(x)−f(−x)
f(x)+f(−x)
e3−2x⋅tan2x
A
[x]−x
B
e3−2x⋅tan2x
C
f(x)+f(−x)
D
f(x)−f(−x)
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Solution
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Let f(x) be any function.
Therefore
If f(x) is an odd function
h(x)=f(x)+f(−x)
=0 ...(i)
If f(x) is an even function
h(x)=f(x)+f(−x)
h(x)=2f(x) ...(ii)
From i and ii we see that h(x) is always an even function.
Hence f(x)+f(−x) will be even function.
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