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quarto/precalc/inversefunctions.qmd
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A (univariate) mathematical function relates or associates values of $x$ to values $y$ using the notation $y=f(x)$. A key point is a given $x$ is associated with just one $y$ value, though a given $y$ value may be associated with several different $x$ values. (Graphically, this is the vertical line test.)
We may conceptualize such a relation in many ways: through an algebraic rule; through the graph of $f;$ through a description of what $f$ does; or through a table of paired values, say. For the moment, let's consider a function as rule that takes in a value of $x$ and outputs a value $y$. If a rule is given defining the function, the computation of $y$ is straightforward. A different question is not so easy: for a given value $y$ what value - or *values* - of $x$ (if any) produce an output of $y$? That is, what $x$ value(s) satisfy $f(x)=y$?
We may conceptualize such a relation in many ways: through an algebraic rule; through the graph of $f;$ through a description of what $f$ does; or through a table of paired values, say. For the moment, let's consider a function as a rule that takes in a value of $x$ and outputs a value $y$. If a rule is given defining the function, the computation of $y$ is straightforward. A different question is not so easy: for a given value $y$ what value - or *values* - of $x$ (if any) produce an output of $y$? That is, what $x$ value(s) satisfy $f(x)=y$?
*If* for each $y$ in some set of values there is just one $x$ value, then this operation associates to each value $y$ a single value $x$, so it too is a function. When that is the case we call this an *inverse* function.