Question

if ( f(-x)=-f(x) ), then a function is odd. explain this relationship in your own words using what you learned about reflections of the graph.

Explanation:

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.