How would you use set builder notation to show that sine is discontinuous at every interval pi?
Correction: that a FUNCTION is discontinuous at every interval pi because sineX is the denominator and x cannot equal 0
To use set builder notation to show that sine is discontinuous at every interval of pi, you would express the set of points where the function is discontinuous.
In this case, we want to show that sine is discontinuous at every interval of pi. In other words, we want to express the set of x-values where sine has a jump discontinuity or removable discontinuity every pi units.
We can use set builder notation to represent this set. The notation can be written as:
{x | x = n * pi, where n is an integer}
Let's break down this notation:
- The variable "x" represents the input values of sine.
- The vertical bar "|" indicates "such that."
- "x = n * pi" states that x is equal to an integer multiple of pi. This means that x can take on values such as pi, 2pi, 3pi, and so on.
So, the set {x | x = n * pi, where n is an integer} represents the set of x-values where sine is discontinuous at every interval of pi.
Notice that for each integer n, the value of x = n * pi represents the endpoints of each interval. At these points, sine has jump discontinuities (where the function "jumps" from one value to another) or removable discontinuities (where the function has a hole or gap).
By using set builder notation, we have expressed the set of points where sine is discontinuous in a concise and mathematical way.