Let's analyze the function [tex]\( y = 150x \)[/tex], which represents the number of calories [tex]\( y \)[/tex] consumed after eating [tex]\( x \)[/tex] pieces of candy. In this function:
- [tex]\( y \)[/tex] is the number of calories.
- [tex]\( x \)[/tex] is the number of pieces of candy consumed.
### Domain of the Function:
The domain of a function refers to all possible values that the independent variable [tex]\( x \)[/tex] can take. In this context, [tex]\( x \)[/tex] represents the number of pieces of candy, which can be any real number because candies are hypothetical and could be broken into smaller pieces. Therefore, [tex]\( x \)[/tex] is not restricted to just integer values and can take any real number value.
Thus, the domain of the function [tex]\( y = 150x \)[/tex] is:
[tex]\[ \text{Domain} = \mathbb{R} \quad (\text{All real numbers}) \][/tex]
### Determining if the Function is Discrete or Continuous:
A function is said to be discrete if its domain consists of isolated points. Conversely, a function is considered continuous if it can take on any value within an interval and there are no interruptions or gaps in the values of the function.
In the context of the function [tex]\( y = 150x \)[/tex]:
- Since [tex]\( x \)[/tex] can take any real number, this allows for an infinite number of possibilities without any gaps or interruptions.
- Additionally, the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is linear and smooth, which indicates there are no breaks or jumps in the values of the function.
Therefore, the function [tex]\( y = 150x \)[/tex] is continuous.
### Conclusion:
- Domain: The domain of the function [tex]\( y = 150x \)[/tex] is all real numbers, [tex]\(\mathbb{R}\)[/tex].
- Discrete or Continuous: The function is continuous because it can take any real value without interruption.
So, the domain of the function is [tex]\(\mathbb{R}\)[/tex] and the function is continuous.
