The teacher's hypothesis is horribly inaccurate. First of all, Scenario A is the only linear function in the group consisting of A,B, and C. Scenario B is a function, but not linear. Scenario C is not a function. Scenario A has all the criteria of a linear function. For every independent variable (aka “x” value or input) in the domain, there is one and only one dependent variable (aka output or “y” value) in the range. It can be written in the form “y=mx+b” where “m” and “b” are real numbers, “x” is an independent variable, and “y” is a dependent variable. It can be shown as a table with a unique value of “y” for every value of “x”. It graphs as a line. Therefore, Scenario A is linear and a function. Scenario B is a function because there is one and only one value of “y” (aka dependent variable or output) for every value of “x” (aka independent variable or input). It can be written as a table with corresponding domain and range values. It cannot be written in linear form “y=mx+b” because “x²” is not a real number, but a variable in the equation “y=10x-x²”. It does not graph as a diagonal line, but as a parabola. Hence, it is only a function. In Scenario C, “x=y²” is not a function and not linear. It is not a function because for some of the “x” values (aka inputs or independent variables) there is more than one “y” value (aka output or dependent variable.) It is not linear because it cannot be written in the form “ y=mx+b” and does not graph as a line. Thus, it is neither linear nor a function. With my facts supporting me, I have come to the conclusion that just because a scenario has “x” and “y” values, it does not necessarily mean it's a linear function.
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