Suppose $f$ and $g$ are functions, with domains $D_f$ and $D_g$,
$f: D_f \rightarrow R$, and $g: D_g \rightarrow R$.
Definition:We say that $g$ is an inverse function
for $f$ if (and only if)
(i) for every $x \in D_f, f(x) \in D_g$
, and $g(f(x))=x$, and (ii) for every $x \in D_g, g(x) \in D_f$ , and
$f(g(x))=x$.
Facts:
If $g$ is an inverse function for $f$ and
$\hat g$ is an inverse function for $f$ then $g = \hat g$.
A function $f$ has an inverse function $g$ if
and only if $f$ is one to one with $D_g = \{ y \in R : y= f(x)$
for some $ x \in D_f\}$
DiagramINV.0Mapping
Diagram for $g$ as inverse of $f$ showing $f(g(x))=x$
ExampleINV.REFL.0 Suppose $f(x) = -2x + 1$.
Verify that $g(x) = -\frac {x-1} 2$ is the inverse function for $f$.
