if ( f(-x)=-f(x) ), then a function is odd. explain this relationship in your own words using what you…

Brief Explanations:

When a function (y = f(x)) is odd ((f(-x)=-f(x))), its graph has symmetry about the origin. Reflecting the graph of (y = f(x)) over the (y) - axis (which gives (y = f(-x))) and then reflecting the resulting graph over the (x) - axis (which gives (y=-f(-x))) results in the original graph (y = f(x)). In other words, if we take a point ((x,y)) on the graph of (y = f(x)), the point ((-x,-y)) (which is obtained by first reflecting ((x,y)) over the (y) - axis to get ((-x,y)) and then reflecting ((-x,y)) over the (x) - axis to get ((-x,-y))) is also on the graph. This is the geometric interpretation of the algebraic property (f(-x)=-f(x)) for an odd function.

Answer:

An odd function (f(x)) has a graph that is symmetric about the origin. Algebraically, (f(-x)=-f(x)) means that for every (x) in the domain of (f), if we first reflect the point ((x,f(x))) over the (y) - axis (to get ((-x,f(x)))) and then over the (x) - axis (to get ((-x,-f(x)))), we land on another point ((-x,f(-x))) on the graph of (f). So the graph of an odd function has symmetry with respect to the origin.