Question

Odd extension is obtained by replacing x by (-x) in the equation of f(x).

• A. True
• B. False

Answer 1

The correct option is B

False

In case of an even extension what we do is we extend the domain on the other side of X- axis, I.e if the function is defined for positive x-axis we extend it to negative x –axis and vice versa. The property of an even function is that it gives you the same value for x as well as for (-x).

So if we replace x by –x in the expression of f(x) and we put the other part of x values I.e. negative of x values we will get the same output.

Example $-f\left(x\right)=x^2+2x,x\ge 0$

Now to extend the domain for x < 0 in such a way that it becomes even.

We'll replace x by –x

So, $f(-x)=(-x{)}^2-2x,x<0$

Now, you see if we put the negative values of x, wouldn't we get the same output which we were getting on putting the positive values of x in the first expression. Yes, we would.

Now to make it an odd function we want to get the output exactly equal but opposite in sign from what we were getting in first expression. So we'll multiply the $2^{nd}$ expression by (-1).

So for odd extension,

$x^2+2x,x\ge 0$

f(x) =

$-x^2+2x,x<0$