Carlos and Ethan noticed that both proportional relationships and linear functions form a straight line when graphed. Carlos claims that all linear functions are also proportional relationships. Ethan disagreed and tells Carlos that he has it backwards, that all proportional relationships are linear functions. Which statement accurately compares their observations? A) Carlos is correct because all linear functions go through the origin.
B) Ethan is correct because all proportional relationships form a straight line and go through the origin and linear functions are linear, but they don’t all go through the origin so they are not always proportional.
C) Carlos is incorrect because no proportional relationships are linear.
D) Both Carlos and Ethan are incorrect, no proportional relationships are nonlinear.

B. Ethan is correct because all proportional relationships form a straight line and go through the origin and linear functions are linear, but they don’t all go through the origin so they are not always proportional.

Step-by-step explanation:

So a proportional relationship is just a special kind of linear relationship, i.e., all proportional relationships are linear relationships (although not all linear relationships are proportional).

the range of a function is the range of y-values for which it is defined, or exists. given that we are only looking at the y-values, we can already discount the first two answers as they deal with the domain of the function (range of x-values).

now, what we need to do is find the smallest and highest y-values for which the function is defined. if you'd like, you could take a ruler and start at the very bottom of the graph, find the lowest point, then move the ruler higher and higher until you reach the highest point. doing so either in your head or with an actual ruler, we can see that the lowest y-value is -5 and the highest y-value is -1.

the question now remains as to whether they are included in the range (fourth answer) or excluded (third answer). the key here is to recognise whether they are marked with an open or a closed circle. since both (-2, -5) and (2, -1) are marked with a closed circle, they are points that exist on the graph and are thus included in the range (note that if they were marked with an open circle like (4, -2) then that would signify that there is a break in the graph at that point).

thus, the fourth answer, (y | -5 ≤ y ≤ -1), is correct.

also note that usually it is good practice to scan your eye inbetween the highest and lowest point for any breaks in the graph - we do have one here at (4, -2), however -2 is still included in the range because there is another point that is included in the graph with the same y-value, ie. point (1, -2).

i hope that but if you require any further explanation don't hesitate to comment below.

ye

step-by-step explanation: