Understanding Even & Odd Functions: Questions Answered

[tina_singh]

1)-why is x(t)+x(-t) always even??..no matter if x(t) even or odd?

2)-when we talk about unit step function...u(t)..and we add..u(t)+u(-t)..the value of both is 1 at t=0..so does'n't that gets added twice??..and it becomes 2 at t=0...

3)when we have x(-t) and we time shift it say x(-t-3) it shifts toward the -ve t axis.. where as x(t-3) the function is shifted on the + axis..why is it so??

i would be really greatful if you can help me out with the above 3 doubts..

 

[micromass]

What did you try to answer this??

For the first, fill in -a in x(t)+x(-t) and see if

$$x(-a)+x(-(-a))=x(a)+x(-a)$$

For the second one. It actually never matters what the unit step function is in 0. So saying that u(t)+u(-t)=2 in 0, is correct, but it doesn't matter.
Note that a lot of people choose that the unit step function is 0, or 1/2 in 0.

The third one. We actually have a function y(t)=x(-t). Then x(-t-3)=y(t+3). So it makes sense that it gets shifted to the other side.

 

[tina_singh]

okh..thanks for answering..i got the first 2 parts...
buh i m still a little confuse about the third...see when we say there are 2 functions x(t) and x(t-2) it means x(t-2) is delayed by 2 sec with respect to x(t) therefore it shifts in the positive x direction.. does'n't the same apply for x(-t) and x(-t-2) the second function is time delayed by 2 secs with respect to the first so even it should shift to the positive x axis..isn't it?

 

[micromass]

The - in front of the t reverses the direction. So shifting towards the positive axis becomes shifting towards the negative axis and vice versa.

 

[CellCoree]

1) The sum of two even functions or two odd functions will always result in an even function. This is because an even function is symmetrical about the y-axis, meaning that it is unchanged when x is replaced with -x. Similarly, an odd function is symmetrical about the origin, meaning that it is unchanged when both x and y are replaced with -x and -y. When we add an even function x(t) with an odd function x(-t), the result will be symmetrical about the y-axis and therefore an even function.

2) The unit step function u(t) is defined as 0 for t < 0 and 1 for t ≥ 0. When we add u(t) and u(-t), we are essentially adding two functions that are both 1 at t = 0. However, we must remember that the unit step function is only defined for t ≥ 0. Therefore, at t = 0, the function u(-t) is undefined and is not included in the addition. This means that the result will still be 1 at t = 0, and not 2.

3) When we time shift a function, we are essentially shifting the entire function along the x-axis. So when we have x(-t) and we shift it by -3, we are shifting the function towards the negative x-axis. On the other hand, when we have x(t-3) and we shift it by -3, the function is shifted towards the positive x-axis. This is because the negative sign in the time shift represents a shift in the opposite direction of the x-axis. So in the first case, the function is shifted towards the negative x-axis, and in the second case, it is shifted towards the positive x-axis.