Domain
The domain of a Function is the Set of inputs the function is allowed to accept. If , then is the domain.
The domain is part of the function, not an afterthought. The same formula can define different functions when the domain changes. For example,
\quad f(x)=x^2$$ is not [[Injective function|injective]], while $$g:[0,\infty)\to[0,\infty), \quad g(x)=x^2$$ is [[Bijective function|bijective]] and has inverse $g^{-1}(y)=\sqrt y$. ## Natural domain The natural domain of a formula is the largest set where the formula makes sense, given the intended number system. For real-valued functions: - $1/x$ excludes $x=0$. - $\sqrt{x}$ requires $x\ge0$. - $\ln x$ requires $x>0$. But in applications the domain may be smaller than the natural domain. A time variable in a lab run might be restricted to an [[Interval]] $[0,T]$, and a radius is usually restricted to $[0,\infty)$. ## Physics link For position as a function of time, $x:I\to\mathbb R^3$, the domain $I$ is the time interval of the model. Its [[Derivative]] gives [[Velocity]] only at times where the derivative exists. Related: [[Codomain and range]], [[Function]], [[Inverse functions]].