1.7 Function notation

Function notation

Functions are represented in math by parentheses. When you write $f(x)$ you indicate that the variable $f$ is a function of—or depends on—the variable $x$ .

For instance, suppose $f(x)=x^2+\mathrm{3x}$ . This means that f is a function that takes whatever you give it, and squares it, and multiplies it by 3, and adds those two quantities.

$\begin{array}{c}7\to \\ \text{10}\to \\ x\to \\ y\to \\ \text{a dog}\to \end{array}$

$\begin{array}{c}\to f(7)=7^2+3(7)=\text{70}\\ \to f(\text{10})={\text{10}}^2+3(\text{10})=\text{130}\\ \to f(x)=x^2+\mathrm{3x}\\ \to f(y)=y^2+\mathrm{3y}\\ \begin{array}{}\to f(\text{dog})={(\text{dog})}^2+3(\text{dog})\\ (\text{*not in the domain})\end{array}\end{array}$

The notation $f(7)$ means “plug the number 7 into the function $f$ .” It does not indicate that you are multiplying $f$ times 7. To evaluate $f(7)$ you take the function $f(x)$ and replace all occurrences of the variable x with the number 7. If this function is given a 7 it will come out with a 70.

If we write $f(y)=y^2+\mathrm{3y}$ we have not specified a different function . Remember, the function is not the variables or the numbers, it is the process. $f(y)=y^2+\mathrm{3y}$ also means “whatever number comes in, square it, multiply it by 3, and add those two quantities.” So it is a different way of writing the same function.

Just as many students expect all variables to be named $x$ , many students—and an unfortunate number of parents—expect all functions to be named $f$ . The correct rule is that—whenever possible— functions, like variables, should be named descriptively . For instance, if Alice makes $100/day, we might write:

• Let m equal the amount of money Alice has made (measured in dollars)
• Let t equal the amount of time Alice has worked (measured in days)
• Then, $m(t)=\text{100}t$

This last equation should be read “ $m$ is a function of $t$ (or $m$ depends on $t$ ). Given any value of the variable $t$ , you can multiply it by 100 to find the corresponding value of the variable $m$ .”

Of course, this is a very simple function! While simple examples are helpful to illustrate the concept, it is important to realize that very complicated functions are also used to model real world relationships. For instance, in Einstein’s Special Theory of Relativity, if an object is going very fast, its mass is multiplied by $\frac{1}{\sqrt{1-\frac{v^2}{9\cdot {\text{10}}^{\text{16}}}}}$ . While this can look extremely intimidating, it is just another function. The speed $v$ is the independent variable, and the mass $m$ is dependent. Given any speed $v$ you can determine how much the mass $m$ is multiplied by.