# Inverse functions

## Example: an inverse function in a population model

• Recall the exponential model for the population of the Earth:
• $$f(t)=(1436.53)\cdot(1.01395)^t$$, where $$t$$ is the number of years since $$1900$$, and $$P=f(t)$$ is the population measured in millions of people.
• We saw that it can be used to predict the population at eg. 2020: $$f(t=120)\approx7573.549$$.
• Note that $$f(t)$$ assigns years $$t$$ to population values $$P$$.
• Suppose you want to predict when the population will exceed 8 billion.
• This means that you want to reverse the assignment, and assign to population values $$P$$ the years $$t$$.
• This can be done via the inverse function $$t=f^{-1}(P)$$ of $$f(t)$$.
• The prediction will be $$f^{-1}(P=8000)\approx123.954$$, that is the model predicts that population will exceed 8 billion in 2023.

## One-to-one functions

• Recall that a function $$y=f(x)$$ assigns precisely one value $$f(x)$$ to every $$x$$ in its domain.
• This means that in order for the inverse $$x=f^{-1}(y)$$ to be a function, for each $$y$$, there can be at most one $$x$$ such that $$f(x)=y$$.
• Definition. A function $$f(x)$$ is called one-to-one, if it never takes on the same value twice: $$f(x_1)\ne f(x_2)$$ whenever $$x_1\ne x_2$$.
• For example, the function $$f(x)=x^2$$ is not one-to-one, because we have $$f(x=-1)=(-1)^2=1=1^2=f(x=1)$$.
• On the other hand, the function $$f(x)=x^3$$ is one-to-one.
• In the next week, we will learn about continuous functions. In particular, we will learn the Intermediate Value Theorem, which will imply that if the domain $$D$$ of a continuous function $$f(x)$$ is one connected interval, with its endpoints possibly $$\pm\infty$$, then it's one-to-one precisely when it's strictly monotonous.
• Horizontal line test. A function is one-to-one if and only if no horizontal line intersects its graph more than once.
• Exercises: 1.5: 10,12

## Definition of inverse functions

• Definition. Let $$y=f(x)$$ be a function with domain $$D$$ and range $$R$$. Then its inverse function $$x=f^{-1}(y)$$ has domain $$R$$ and range $$D$$, and it is defined by $$$\tag{*} f^{-1}(y)=x\text{ if and only if }y=f(x)\text{ for any y in R}.$$$
• Caution. Do not confuse the inverse function $$f^{-1}(x)$$ with the reciprocal $$f(x)^{-1}=\frac{1}{f(x)}$$. Note that in the inverse function, the $${}^{-1}$$ is after the function name, and in the reciprocal, it is after the argument.
• Cancellation equations. By substituting the first equation in the definition (*) to the second and vice versa, we get the following. \begin{align*} y=f(f^{-1}(y))&\text{ for any y in R}\\ f^{-1}(f(x))=x&\text{ for any x in D} \end{align*}

## Finding inverse functions

• Let $$f(x)$$ be a one-to-one function with range $$R$$.
• To find its inverse function $$f^{-1}(y)$$, you need to solve the equation $$y=f(x)$$ for $$x$$ for every $$y$$ in $$R$$.
• Afterwards, you can interchange $$x$$ and $$y$$ to get the inverse in the form $$f^{-1}(x)$$, if you want to.
• Exercises: Find the inverses of $$2x-3$$, $$x^3+2$$, and $$x^2-x$$, the latter with domain $$x\ge\frac12$$.
• Click here for the graphs of the functions and their inverses.
• Note that the graph of the inverse $$y=f^{-1}(x)$$ is obtained by mirroring the graph of the original function $$y=f(x)$$ with respect to the line $$y=x$$.
• Exercises: 1.5.22,20