Exponential functions

Summary

• Given: a real-valued function of a real variable, \(f: R→R\)
• We say that \(f\) is a exponential function if there is a constants \(a>0\) such that

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

• for all \(x\) in the domain
• \(f\) is strictly increasing if \(a>1\) and strictly decreasing if \(0 < a < 1\)
• The range of \(f\) is given by \(\left( 0,∞ \right)\)
• If \(f\) is an exponential function then \(f(x+1)=af(x) \) for all \(x\) . \(y=f(x)\) increases by \(\left( a-1 \right)⋅100%\) when \(x\) increases by one unit.
• If \(f(x)=e^x\) where \(e=2.7182…\) then \(f\) is the natural exponential function. \(e^x\) and \(exp \left( x \right)\) means the same thing.
• If \(f\left( x \right)=a^{bx}\) for a constant \(b\) we also call it an exponential function.