Question

Argh with this piecewise functions! Please help...
f(x) = { csc (PI x / 6), |x-3| <= 2
{ 2, |x-3| > 2

Answer 1

What must we do with it?

Answer 2

The x-3 is messing me up.

Answer 3

Actually, its the fact that its a trig function that's really making it difficult to break apart for me.

Answer 4

Oops, sorry. We are supposed to find the x-values (if any) at which the function is discontinuous. And state if it is removable or not.

Answer 5

The \(|x-3| > 2\) is basically saying that \(x\) can be any number between \(1<x<5\). So if \(x\) is between 1 and 5, \(f(x) = 2\), otherwise \(f(x) = csc (\frac{\pi x}{6})\)

Answer 6

I got to the 1 and 5 part then got stuck. :)

Answer 7

The graphs of the 2 functions show that they do not meet anyway. But I am trying to figure out how to explain that algebraically and I am failing.

Answer 8

My answer didn't seem to post, but you have to take the left/right limits and see if the equal the function at those points. For example:
\[\lim_{x \to 1^-} f(x)= \lim_{x \to 1^-} \csc \left ( \frac{\pi x}{6} \right ) = \csc \left ( \frac{\pi}{6} \right ) = 2 \]
and
\[\lim_{x \to 1^+ }f(x) = \lim_{x \to 1^+} 2 = 2 = \lim_{x\to 1^-} f(x) = f(2) \]
So it is continuous at x = 1. Be sure to remember to check the dicontinuities of csc(x)

Answer 9

Wait, let me digest this according to the given problem.

Answer 10

I tried graphing the functions and at no point do they meet. But it seems like at x=3 is where they might meet. Is this pertinent?

Answer 11

They should meet:
http://www.wolframalpha.com/input/?i=plot+y+%3D+csc%28pi*x%2F6%29+and+y+%3D+2

Answer 12

Not quite. At x =3, the trig function is at .0998
With the linear function at x=3, f(x) = 0

Answer 13

wait, you're right.

Answer 14

I'm still confused

Answer 15

For a piecewise function, you must make sure the left and right hand limits match so to make sure the limit exists. Make sure the point exists there too. If these are true, then the function is continuous.

But be aware: discontinuities can exist elsewhere (namely csc(x) has MANY discontinuities)

Answer 16

I understand that, however, the conditions with the absolute value of x-3 is confusing me.

Answer 17

I'm checking out your graph to try to make sense of the conditions...

Answer 18

Ah, remember how you said you got to the x is between 1 and 5. We just need to check the "end points" of the interval.

Answer 19

By check, I mean take the limits and check to see if continuity conditions hold

Answer 20

Oh ok. But, does that mean I am plugging in both values for x?

Answer 21

Yes, you must check 1 (I posted that as an example above) and x = 5. After that, check to see where csc(pix/6) is not continuous.

Answer 22

Ok, thanks so much. I'll try it.