Functions

Aug 2008
184
6
Let \(\displaystyle f_1(x)\) and \(\displaystyle f_2(x)\) be odd and even functions respectively. How can we construct an even function out of these?
 
Jan 2010
354
173
Plenty of ways, but why can't you just take \(\displaystyle f_2(x)\) as the even function? What's the point of using \(\displaystyle f_1(x)\) in constructing an even function?

Edit: In case you mean you need to construct an even function from only \(\displaystyle f_1(x)\), some simple ways would be to take the absolute value or square the function.

\(\displaystyle g(x)=|f_1(x)|\)

\(\displaystyle h(x)=(f_1(x))^2\)

These would both be even functions.
 
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HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
What does "construct out of them" mean?
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
Plenty of ways, but why can't you just take \(\displaystyle f_2(x)\) as the even function? What's the point of using \(\displaystyle f_1(x)\) in constructing an even function?

Edit: In case you mean you need to construct an even function from only \(\displaystyle f_1(x)\), some simple ways would be to take the absolute value or square the function.

\(\displaystyle g(x)=|f_1(x)|\)

\(\displaystyle h(x)=(f_1(x))^2\)

These would both be even functions.
And if you really have to use both functions, so would \(\displaystyle |f_1(x)|+ f_2(x)\) and \(\displaystyle (f_1(x))^2+ f_2(x)\).

Now, if the problem had been to construct an even function from two odd functions, that would have been a little more interesting!
 
Jun 2010
89
14
Iran
Also their composition{ g(x) = f1(f2(x)) and h(x) = f2(f1(x)) } are even!