Question 181302
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The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a <i><b>single output</b></i> to each input element drawn from a fixed set, such as the real numbers, although different inputs may have the same output.


This definition leads directly to what is called the Vertical Line Test.  If you graph a mathematical relationship in *[tex \Large R^2] and you can find <i><b>any</b></i> vertical line that intersects the graph in more than one point, then the relation is <i><b>not</b></i> a function.


All two-variable linear equations graph to straight lines, so the only graphs that would be intersected in more than one point by a vertical line would be the graph of a vertical line.  Hence, all linear equations <i><b>except</b></i> for the sub-set of linear equations of the form *[tex \Large x = a] are functions.  An equation of the form *[tex \Large x = a] would be intersected by the vertical line *[tex \Large x = a] in <i><b>all</b></i> of its points, causing the Vertical Line Test to fail.


So, the answer to your initial question is no.  All of the instances where a linear equation mapped to *[tex \Large R^2] is <i><b>not</b></i> a function is *[tex \Large \{Ax + By = C | x,\  y,\  A, \ C\  \in \R,\  A \neq 0,\  B = 0\}].


A non-linear equation is anything that doesn't graph to a straight line.  The equation of a non-linear <b>function</b> must also pass the Vertical Line Test.  Common examples are polynomial equations of degree <i>n</i> where *[tex \Large n > 1], exponential equations such as *[tex \Large b^x = y], or logarithmic equations such as *[tex \Large \log_b{x} = y].



John
*[tex \LARGE e^{i\pi} + 1 = 0]
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