# Changes

| [[scalar multiple of function|scalar multiple]] by a constant || $af$ is the function $x \mapsto af(x)$ where $a$ is a real number || Same as the domain of the original function ||
| [[pointwise product of functions|pointwise product]] || $f \cdot g$ (sometimes denoted $fg$) is the function $x \mapsto f(x)g(x)$<br>f_1 \cdot f_2 \cdot \dots f_n[/itex] (sometimes denoted $f_1f_2\dots f_n$ is the function $x \mapsto f_1(x)f_2(x) \dots f_n(x)$ || Intersection of the domains of all the functions being multiplied || Similar to the note for sums.<br> <toggledisplay>Note that this definition is quasi-paradoxical, in that the function defined as $f \cdot g$ where $f(x) = x - 2$ and $g(x) = 1/(x - 2)$ is 1 everywhere ''except'' at 2. This is because even though the expressions for $f$ and $g$ cancel each other, evaluating at $x = 2$ does not make sense for the expression ''prior'' to algebraic simplication by cancellation.</toggledisplay>|-| [[pointwise quotient of functions|pointwise quotient]] || $f/g$ is the function $x \mapsto f(x)/g(x)$ || Intersection of the domain of $f$ with the ''subset'' of the domain of $g$ comprising those points $x$ where $g(x) \ne 0$.|-| [[composite of two functions]] || $f \circ g$ is the function $x \mapsto f(g(x))$ || Set of those values $x$ for which $g(x)$ lies inside the domain of $f$. |||-| [[inverse function]] of a [[one-one function]] || $f^{-1}$ sends <matH>x[/itex] to the unique $y$ such that $f(y) = x$ || Same as the [[range]] of the original one-one function $f$ |||-| [[piecewise definition of functions|piecewise definition]] || {{fillin}} || Union of the domain of definition for each piece. This domain is usually given explicitly or requires finding the subset of an explicitly specified set where an explicit expression makes sense. ||