1. ## Functions

Let $f_1(x)$ and $f_2(x)$ be odd and even functions respectively. How can we construct an even function out of these?

2. Plenty of ways, but why can't you just take $f_2(x)$ as the even function? What's the point of using $f_1(x)$ in constructing an even function?

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

$g(x)=|f_1(x)|$

$h(x)=(f_1(x))^2$

These would both be even functions.

3. What does "construct out of them" mean?

4. Originally Posted by drumist
Plenty of ways, but why can't you just take $f_2(x)$ as the even function? What's the point of using $f_1(x)$ in constructing an even function?

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

$g(x)=|f_1(x)|$

$h(x)=(f_1(x))^2$

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

Now, if the problem had been to construct an even function from two odd functions, that would have been a little more interesting!

5. Also their composition{ g(x) = f1(f2(x)) and h(x) = f2(f1(x)) } are even!