Question 181302: Are all linear equations functions?I would say no but In what instance is a linear equation not a function?I need an equation of nonlinear function and I don't knwo how to write that or where to start. Thank you!
Found 2 solutions by eperette, solver91311:
Answer by eperette(173)   (Show Source):
Answer by solver91311(24713)  (Show Source):
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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 single output 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  and you can find any vertical line that intersects the graph in more than one point, then the relation is not 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 except for the sub-set of linear equations of the form  are functions. An equation of the form would be intersected by the vertical line in all 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 is not a function is  .
A non-linear equation is anything that doesn't graph to a straight line. The equation of a non-linear function must also pass the Vertical Line Test. Common examples are polynomial equations of degree n where  , exponential equations such as  , or logarithmic equations such as  .
John
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