Discontinuous split function problem

[ucdawg12]

im not really sure how to do this one because I have forgotten most of the things about trig functions from last year, but here is the problem:
(im going to say Ħ=pi because I can't find the character for pi)
f(x)= tan((Ħx)/4) when |x|<1
x when |x| =>(greater than or equal to) 1

Im supposed to find the x values, if any, at which f is not continuous, and find which of the discontinuities are removable.

I haven't had a lot of problems with my split function questions, I just don't know how to handle it when trig functions are involved, can anyone explain?

Thanks

 

[HallsofIvy]

First of all, you can create a π by using & p i ; but without the spaces- it doesn't look like a very good "pi" to me- the other Greek letters using & ...; are better!

Secondly, the way to handle a "piece-wise" function (what you are calling a "split" function) is to look at the individual "pieces" and then look carefully at the point where they meet.

-1< x< 1 f(x)= tan(π x) Okay, where is tan(π x) NOT continuous. That's really the same as asking "where is tan(&pi x) not defined. All "elementary" functions are continuous wherever they are defined.

For x< -1 or x> 1, f(x)= 1. That's easy, that's a constant and so it is always defined and always constant.

Now, what about x= -1 or x= 1? as x-> 1, πx-> &pi. What is the value of tan(π)? As x-> 1, 0-> 0, of course, so that limit is 0. IF limit tan(πx)= tan(π)= 0 then the function is continuous at x= 1.

As x-> -1, π x-> -&pi. What is the value of tan(-π)? As x-> 0 0-> 0, of course, so that limit is 0. IF limit tan(-πx)= tan(π)= 0, then the function is continuous at x= -1.

 

[M98Ranger]

for reaching out for help with this problem. The first step in solving this problem would be to understand what a discontinuous split function is. A discontinuous split function is a function that is defined differently for different intervals of its domain. In this case, the function is defined as f(x)= tan((Ħx)/4) when |x|<1 and x when |x| >= 1. This means that for x values less than 1, the function will be defined by the trigonometric function, and for x values greater than or equal to 1, the function will be defined by the linear function x.

To find the x values where the function is not continuous, we need to look at the points where the two definitions of the function meet. In this case, the two definitions meet at x=1. This means that the function is not continuous at x=1.

Now, to determine if this discontinuity is removable or not, we need to evaluate the limit of the function as x approaches 1 from both the left and the right. If the two limits are equal, then the discontinuity is removable. If the two limits are not equal, then the discontinuity is not removable.

In this case, the limit of the function as x approaches 1 from the left is tan((Ħx)/4)=tan((Ħ(1))/4)=tan(Ħ/4). The limit of the function as x approaches 1 from the right is x=1. Since these two limits are not equal, the discontinuity at x=1 is not removable.

I hope this explanation helps you to better understand how to handle discontinuous split functions involving trigonometric functions. Remember to always evaluate the limits and check for continuity at the points where the definitions of the function meet.