Question about modular functions

[Taturana]

Suppose I have any modular function, for example:

$$f(x) = |2x + 4| + 3$$

I can rewrite the function in the following way:

$$f(x) = \left\{\begin{matrix} 2x + 7, \;\; x \geq -2\\ -2x -1, \;\; x < -2 \end{matrix}\right.$$

right?

Okay, the question is: how do I know that the first condition is $$\geq$$ and second condition is < and not vice-versa?

Thank you,
Rafael Andreatta

 

[Petr Mugver]

It doesn't matter, because |x| is a continuous function. What's the value of f(x) for x=-2?

 

[Taturana]

It doesn't matter, because |x| is a continuous function. What's the value of f(x) for x=-2?

1. You mean, because |x| is a continuous function or because f is a continuous function?

2. What if it was not continuous?

 

[Office_Shredder]

f(x) is a continuous function, so the two halves of the definition must agree where they meet up.

Since adding and multiplying and composing continuous functions is a continuous function, and sum of absolute value functions like f(x) here will be continuous. In the event you're doing something like dividing by the absolute value of x, then you can just:
plug in the value of x for which you're unsure and compare it to your formulae. See which one it agrees with. If the function isn't defined for that value of x, then you don't need to decide which inequality gets the equal sign because you aren't defining the function anyway

 

[CellCoree]

I would like to clarify that modular functions are a type of mathematical function that involves breaking up the input into different intervals and using different equations to describe the function in each interval. In your example, the function is broken into two intervals based on the value of x: x ≥ -2 and x < -2. The first condition, x ≥ -2, is used for values of x that are greater than or equal to -2, while the second condition, x < -2, is used for values of x that are less than -2. This is a convention that is commonly used in modular functions to ensure that each interval is clearly defined and that there are no overlaps. It is important to follow this convention in order to accurately describe the behavior of the function and avoid any confusion.