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[FP2 WJEC] Functions

Hi there,

I am having a little trouble with one question regarding Functions (Even, Odd, Neither). I am unsure how I would approach it, as no specific function is given.

The question is as follows:

Show that;
E(x)=12(f(x)+f(−x)) E\left( x\right) =\dfrac {1}{2}\left( f\left( x\right) +f\left( -x\right) \right)
is an even function.

I thought that I may be able to assume f(x)f\left(x\right) is even, as otherwise the function of E(x)E\left(x\right) would not be even.
Therefore, as f(x)f\left(x\right) is even;

f(x)=f(−x)f\left(x\right)=f\left( -x\right),

and Therefore:

E(x)=12(f(x)+f(x)) E\left( x\right) =\dfrac {1}{2}\left( f\left( x\right) +f\left( x\right) \right)
E(x)=12(2f(x)) E\left( x\right) =\dfrac {1}{2}\left(2 f\left( x\right) \right)
E(x)=f(x) E\left( x\right) =f\left( x\right)

Therefore, since f(x)f\left(x\right) is even, E(x)E\left(x\right) must also be, given they are equal to one another.

This may be completely wrong so any help would be greatly appreciated,

Thanks in advance,
Reply 1
Avatar for TeeEm
TeeEm
19
Original post by Dackor
Hi there,

I am having a little trouble with one question regarding Functions (Even, Odd, Neither). I am unsure how I would approach it, as no specific function is given.

The question is as follows:

Show that;
E(x)=12(f(x)+f(−x)) E\left( x\right) =\dfrac {1}{2}\left( f\left( x\right) +f\left( -x\right) \right)
is an even function.

I thought that I may be able to assume f(x)f\left(x\right) is even, as otherwise the function of E(x)E\left(x\right) would not be even.
Therefore, as f(x)f\left(x\right) is even;

f(x)=f(−x)f\left(x\right)=f\left( -x\right),

and Therefore:

E(x)=12(f(x)+f(x)) E\left( x\right) =\dfrac {1}{2}\left( f\left( x\right) +f\left( x\right) \right)
E(x)=12(2f(x)) E\left( x\right) =\dfrac {1}{2}\left(2 f\left( x\right) \right)
E(x)=f(x) E\left( x\right) =f\left( x\right)

Therefore, since f(x)f\left(x\right) is even, E(x)E\left(x\right) must also be, given they are equal to one another.

This may be completely wrong so any help would be greatly appreciated,

Thanks in advance,


replace x for -x

if f(-x) = f(x) then even
if f(-x) =- f(x) then odd
Reply 2
Avatar for davros
davros
16
Original post by Dackor
Hi there,

I am having a little trouble with one question regarding Functions (Even, Odd, Neither). I am unsure how I would approach it, as no specific function is given.

The question is as follows:

Show that;
E(x)=12(f(x)+f(−x)) E\left( x\right) =\dfrac {1}{2}\left( f\left( x\right) +f\left( -x\right) \right)
is an even function.

I thought that I may be able to assume f(x)f\left(x\right) is even, as otherwise the function of E(x)E\left(x\right) would not be even.
Therefore, as f(x)f\left(x\right) is even;

f(x)=f(−x)f\left(x\right)=f\left( -x\right),

and Therefore:

E(x)=12(f(x)+f(x)) E\left( x\right) =\dfrac {1}{2}\left( f\left( x\right) +f\left( x\right) \right)
E(x)=12(2f(x)) E\left( x\right) =\dfrac {1}{2}\left(2 f\left( x\right) \right)
E(x)=f(x) E\left( x\right) =f\left( x\right)

Therefore, since f(x)f\left(x\right) is even, E(x)E\left(x\right) must also be, given they are equal to one another.

This may be completely wrong so any help would be greatly appreciated,

Thanks in advance,


You can't assume anything about f(x) - that's the point of the question!

You just need to use the definition of "even" for a function.

You want to prove that E(-x) = E(x). Can you do this just from the definition you are given for E(x)?

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