Power functions

Summary

• Given: a real-valued function \(f\) of a real variable
• We say that \(f\) is a power function if there are constants \(r\) such that

\[f\left( x \right)=x^r\]

• for all \(x\) in the domain
• If \(f(x)=x^2\) then \(f\) is a power function as well as a quadratic function.
• If \(f(x)=1/x\) then \(f\) is a power function as well as a rational function.
• The natural domain is:
• \(r\) is a natural number: \(R\)
• \(r\) is zero or a negative integer: \(R∖\{ 0 \}\)
• \(r>0\) , not an integer: \(\left[ 0,∞ \right)\)
• \(r < 0\) , not an integer: \(\left( 0,∞ \right)\)
• \(f\) is strictly increasing if \(r>0\) and strictly decreasing if \(r < 0\)
• If \(f\left( x \right)=cx^r\) for a constant \(c\) we also call it a power function.