# Functions

#### roshanhero

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?

#### drumist

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.

• HallsofIvy

#### HallsofIvy

MHF Helper
What does "construct out of them" mean?

#### HallsofIvy

MHF Helper
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!

#### Mathelogician

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