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2025-07-27 15:26:00 -04:00
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quarto/precalc/inversefunctions.qmd
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# The Inverse of a Function
# The inverse of a function
{{< include ../_common_code.qmd >}}
@@ -25,7 +25,7 @@ We may conceptualize such a relation in many ways:
* 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 $f(x)=y$?
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.
@@ -202,7 +202,7 @@ In the section on the [intermediate value theorem](../limits/intermediate_value_
## Functions which are not always invertible
Consider the function $f(x) = x^2$. The graph--a parabola--is clearly not *monotonic*. Hence no inverse function exists. Yet, we can solve equations $y=x^2$ quite easily: $y=\sqrt{x}$ *or* $y=-\sqrt{x}$. We know the square root undoes the squaring, but we need to be a little more careful to say the square root is the inverse of the squaring function.
Consider the function $f(x) = x^2$. The graph---a parabola---is clearly not *monotonic*. Hence no inverse function exists. Yet, we can solve equations $y=x^2$ quite easily: $y=\sqrt{x}$ *or* $y=-\sqrt{x}$. We know the square root undoes the squaring, but we need to be a little more careful to say the square root is the inverse of the squaring function.
The issue is there are generally *two* possible answers. To avoid this, we might choose to only take the *non-negative* answer. To make this all work as above, we restrict the domain of $f(x)$ and now consider the related function $f(x)=x^2, x \geq 0$. This is now a monotonic function, so will have an inverse function. This is clearly $f^{-1}(x) = \sqrt{x}$. (The $\sqrt{x}$ being defined as the principle square root or the unique *non-negative* answer to $u^2-x=0$.)
@@ -287,7 +287,7 @@ plot(xs, ys; color=:blue, label="f",
plot!(ys, xs; color=:red, label="f⁻¹") # the inverse
```
By flipping around the $x$ and $y$ values in the `plot!` command, we produce the graph of the inverse function--when viewed as a function of $x$. We can see that the domain of the inverse function (in red) is clearly different from that of the function (in blue).
By flipping around the $x$ and $y$ values in the `plot!` command, we produce the graph of the inverse function---when viewed as a function of $x$. We can see that the domain of the inverse function (in red) is clearly different from that of the function (in blue).
The inverse function graph can be viewed as a symmetry of the graph of the function. Flipping the graph for $f(x)$ around the line $y=x$ will produce the graph of the inverse function: Here we see for the graph of $f(x) = x^{1/3}$ and its inverse function: