Describe in detail what the discontinuities that could be if a rational function has a linear function for a numerator and an absolute value function for a denominator.
I don't get this because wouldn't there be an infinite number of discontinuities depending on the transformations of the linear/absolute value?
put a line ovr that number.a line over a number means infinity.
You are correct that the number of discontinuities depends on the specific transformations of the linear and absolute value functions. However, in general, there are certain types of discontinuities that can occur when a rational function has a linear numerator and an absolute value denominator.
Discontinuities in a rational function occur when the denominator is equal to zero, as this would make the function undefined. In the case of a rational function with a linear numerator and an absolute value denominator, the absolute value function can introduce additional ways for the denominator to be zero.
1. Vertical Asymptotes: When the absolute value function in the denominator is of the form |x - a|, there can be a vertical asymptote at x = a. This occurs when the expression inside the absolute value function approaches zero, making the denominator approach zero as well.
2. Holes: Depending on the transformation of the linear function or the absolute value function, there may be values of x at which the numerator and denominator both become zero. In this case, there is a hole in the graph of the rational function. The hole represents a point that is removed from the graph, but can be easily filled by simplifying and canceling out the common factors.
3. Removable Discontinuity: Sometimes, after simplifying the rational expression, you may find a common factor between the numerator and denominator that cancels out and results in a hole. This means that the function is not defined at that point, but you can redefine the function to include that point as a limit.
In summary, while there can be an infinite number of discontinuities depending on the specific transformations, common types of discontinuities in a rational function with a linear numerator and an absolute value denominator include vertical asymptotes, holes, and removable discontinuities. It is important to consider the specific values and transformations to determine the precise locations of these discontinuities.