is x=1 a linear function?

Answer 1

No, the relation x = 1 is not a linear function although its graph in an x-y-coordinate plane is a straight line.

Define the relation R as the set of all (x, y) ordered pairs in the Euclidean plane such that x = 1. Note that the definition of a function requires that if the ordered pair (a, b) is in R and the ordered pair (a, c) is in R, then b must be identical with c. However, for us, we see that (1, 3) is in R and (1, 5) is in R, but 3 is not equal to 5. Thus, R cannot be a function. Because R is not a function and all linear functions are functions, R is not a linear function.

Another way to see this is to use the vertical line test.

Answer 2

Yes, it's a linear relation. It is linear.

but not a function, linear function or function of any kind

a linear relation has degree 1

x has an implicit 1 as the degree. x^1 = x

another test: graph it. IF the graph is a straight line, it's a linear equation and linear relation. x=1 is a vertical straight line. No curves, no bends. Straight as it comes.

but to be a function (linear or otherwise) it has to pass the vertical line test. There cannot be more than one y value for a given x value. if you draw a veritcal line through the graph, it can't intersect more than once.

all straight lines are linear equations and linear relations

and all straight lines are linear functions, except vertical lines. x=1 is a vertical line.