- Identify the linear function model \(y = 11 + 7x\) based on the provided data points.
- Determine the domain of the function, which is given as \(x \geq 0\).
- Recognize the function is strictly increasing, so the minimum value of \(y\) occurs at the minimum value of \(x\) (where \(x = 0\), \(y = 11\)).
- Conclude that the range is all real numbers greater than or equal to \(\boxed{11}\).
Explanation
- Analysis of the Function and Domain
We are given a relationship between the number of cheese sticks \(x\) and the total grams of protein \(y\) consumed. The data points are \((0, 11), (2.5, 28.5), (5, 46),\) and \((7, 60)\). By checking the difference in protein for a change in cheese sticks, we can determine the relationship. The slope \(m\) is the change in protein over the change in cheese sticks: \(m = \frac{28.5 - 11}{2.5 - 0} = \frac{17.5}{2.5} = 7\). The y-intercept is at \(x = 0, y = 11\). Thus, the function is \(y = 11 + 7x\). Since Livia can eat part of a cheese stick, \(x\) represents any non-negative real number, i.e., \(x \geq 0\).
- Determining the Range
The function is defined as \(y = f(x) = 11 + 7x\), where \(x \geq 0\). Because the function is linear with a positive slope of \(7\), it is a strictly increasing function. This means that as \(x\) increases, \(y\) also increases. To find the range, we identify the smallest and largest possible values of \(y\). The smallest value of \(x\) is \(0\), so the smallest value of \(y\) is \(f(0) = 11 + 7(0) = 11\). As \(x\) increases without bound, \(f(x)\) also increases without bound. Therefore, the function \(f(x)\) takes all values greater than or equal to \(11\).
- Conclusion
Based on our analysis, the range consists of all real numbers starting from \(11\) and going to infinity, because \(x\) is a non-negative real number. Comparing this to the given options:
- all real numbers
- all real numbers greater than or equal to 0
- all real numbers greater than or equal to 11
- all integers greater than or equal to 11
The range is clearly all real numbers greater than or equal to \(11\). Thus, the correct choice is "all real numbers greater than or equal to 11".
Examples
This problem illustrates how a linear function can model resource consumption. In real life, such models are used in nutrition to track intake, in logistics to manage fuel consumption relative to distance, or in manufacturing to calculate material costs relative to production quantity. By identifying the function and its domain, one can effectively predict the outcomes (the range) of their activities, helping in planning and decision-making for efficiency.