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We can also give an algebraic expression as the input to a function. For example $\text{\hspace{0.17em}}f\left(a+b\right)\text{\hspace{0.17em}}$ means “first add a and b , and the result is the input for the function f .” The operations must be performed in this order to obtain the correct result.
The notation $\text{\hspace{0.17em}}y=f\left(x\right)\text{\hspace{0.17em}}$ defines a function named $\text{\hspace{0.17em}}f.\text{\hspace{0.17em}}$ This is read as $\text{\hspace{0.17em}}\u201cy\text{\hspace{0.17em}}$ is a function of $\text{\hspace{0.17em}}x.\u201d\text{\hspace{0.17em}}$ The letter $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ represents the input value, or independent variable. The letter $\text{\hspace{0.17em}}y\text{,\hspace{0.17em}}$ or $\text{\hspace{0.17em}}f\left(x\right),\text{\hspace{0.17em}}$ represents the output value, or dependent variable.
Use function notation to represent a function whose input is the name of a month and output is the number of days in that month.
The number of days in a month is a function of the name of the month, so if we name the function $f,$ we write $\text{days}=f(\text{month})$ or $d=f(m).$ The name of the month is the input to a “rule” that associates a specific number (the output) with each input.
For example, $\text{\hspace{0.17em}}f\left(\text{March}\right)=31,\text{\hspace{0.17em}}$ because March has 31 days. The notation $\text{\hspace{0.17em}}d=f\left(m\right)\text{\hspace{0.17em}}$ reminds us that the number of days, $\text{\hspace{0.17em}}d\text{\hspace{0.17em}}$ (the output), is dependent on the name of the month, $\text{\hspace{0.17em}}m\text{\hspace{0.17em}}$ (the input).
A function $\text{\hspace{0.17em}}N=f\left(y\right)\text{\hspace{0.17em}}$ gives the number of police officers, $\text{\hspace{0.17em}}N,\text{\hspace{0.17em}}$ in a town in year $\text{\hspace{0.17em}}y.\text{\hspace{0.17em}}$ What does $\text{\hspace{0.17em}}f\left(2005\right)=300\text{\hspace{0.17em}}$ represent?
When we read $\text{\hspace{0.17em}}f\left(2005\right)=300,\text{\hspace{0.17em}}$ we see that the input year is 2005. The value for the output, the number of police officers $\text{\hspace{0.17em}}\left(N\right),\text{\hspace{0.17em}}$ is 300. Remember, $\text{\hspace{0.17em}}N=f\left(y\right).\text{\hspace{0.17em}}$ The statement $\text{\hspace{0.17em}}f\left(2005\right)=300\text{\hspace{0.17em}}$ tells us that in the year 2005 there were 300 police officers in the town.
Use function notation to express the weight of a pig in pounds as a function of its age in days $\text{\hspace{0.17em}}d\text{.}$
$w=f(d)$
Instead of a notation such as $\text{\hspace{0.17em}}y=f(x),\text{\hspace{0.17em}}$ could we use the same symbol for the output as for the function, such as $\text{\hspace{0.17em}}y=y(x),\text{\hspace{0.17em}}$ meaning “y is a function of x?”
Yes, this is often done, especially in applied subjects that use higher math, such as physics and engineering. However, in exploring math itself we like to maintain a distinction between a function such as $\text{\hspace{0.17em}}f,\text{\hspace{0.17em}}$ which is a rule or procedure, and the output $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ we get by applying $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ to a particular input $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ This is why we usually use notation such as $\text{\hspace{0.17em}}y=f\left(x\right),P=W\left(d\right),\text{\hspace{0.17em}}$ and so on.
A common method of representing functions is in the form of a table. The table rows or columns display the corresponding input and output values. In some cases, these values represent all we know about the relationship; other times, the table provides a few select examples from a more complete relationship.
[link] lists the input number of each month (January = 1, February = 2, and so on) and the output value of the number of days in that month. This information represents all we know about the months and days for a given year (that is not a leap year). Note that, in this table, we define a days-in-a-month function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}D=f\left(m\right)\text{\hspace{0.17em}}$ identifies months by an integer rather than by name.
Month number, $\text{\hspace{0.17em}}m\text{\hspace{0.17em}}$ (input) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
Days in month, $\text{\hspace{0.17em}}D\text{\hspace{0.17em}}$ (output) | 31 | 28 | 31 | 30 | 31 | 30 | 31 | 31 | 30 | 31 | 30 | 31 |
[link] defines a function $\text{\hspace{0.17em}}Q=g\left(n\right).\text{\hspace{0.17em}}$ Remember, this notation tells us that $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ is the name of the function that takes the input $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ and gives the output $\text{\hspace{0.17em}}Q\text{\hspace{0.17em}.}$
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