1.3 Using Function Notation
Once we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into computers. There are various ways of representing functions. A standard function notation is one representation that facilitates working with functions.
To represent “height is a function of age,” we start by identifying the descriptive variables [latex]h[/latex] for height and [latex]a[/latex] for age. The letters [latex]\text{}f,\text{}g,[/latex] and [latex]\text{}h\text{}[/latex] are often used to represent functions just as we use [latex]x,\text{}y,[/latex] and [latex]z[/latex] to represent numbers and [latex]A,\text{}B,[/latex] and [latex]C[/latex] to represent sets.
[latex]\begin{array}{lllll}h\text{ is }f\text{ of }a\hfill & \hfill & \hfill & \hfill & \text{We name the function }f;\text{ height is a function of age}.\hfill \\ h=f\left(a\right)\hfill & \hfill & \hfill & \hfill & \text{We use parentheses to indicate the function input}\text{. }\hfill \\ f\left(a\right)\hfill & \hfill & \hfill & \hfill & \text{We name the function }f;\text{ the expression is read as “}f\text{ of }a\text{.”}\hfill \end{array}[/latex]
Remember, we can use any letter to name the function; the notation [latex]\text{}h\left(a\right)\text{}[/latex] shows us that [latex]h[/latex] depends on [latex]a[/latex]. The value [latex]a[/latex] must be put into the function [latex]h[/latex] to get a result. The parentheses indicate that age is input into the function; they do not indicate multiplication.
We can also give an algebraic expression as the input to a function. For example [latex]\text{}f\left(a+b\right)\text{}[/latex] 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.
Function Notation
The notation [latex]\text{}y=f\left(x\right)\text{}[/latex] defines a function named [latex]\text{}f\text{}[/latex]. This is read as “[latex]\text{}y\text{}[/latex] is a function of [latex]\text{}x\text{}[/latex]“. The letter [latex]\text{}x\text{}[/latex] represents the input value, or independent variable. The letter [latex]\text{}y\text{,}[/latex]or [latex]\text{}f\left(x\right),\text{}[/latex] represents the output value, or dependent variable.
Example 1: Using Function Notation for Days in a Month
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. Assume that the domain does not include leap years.
Analysis
Note that the inputs to a function do not have to be numbers; function inputs can be names of people, labels of geometric objects, or any other element that determines some kind of output. However, most of the functions we will work with in this book will have numbers as inputs and outputs.
Example 2: Interpreting Function Notation
A function [latex]\text{}N=f\left(y\right)\text{}[/latex] gives the number of police officers, [latex]\text{}N\text{}[/latex], in a town in year [latex]\text{}y\text{}[/latex]. What does [latex]\text{}f\left(2005\right)=300\text{}[/latex] represent?
Example 3: Using Function Notation
Use function notation to express the weight of a pig in pounds as a function of its age in days [latex]\text{}d\text{}[/latex].
Question & Answer
Instead of a notation such as [latex]\text{}y=f\left(x\right)\text{}[/latex], could we use the same symbol for the output as for the function, such as [latex]\text{}y=y\left(x\right)\text{}[/latex], 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 [latex]\text{}f\text{}[/latex], which is a rule or procedure, and the output [latex]\text{}y\text{}[/latex] we get by applying [latex]\text{}f\text{}[/latex] to a particular input [latex]\text{}x\text{}[/latex]. This is why we usually use notation such as [latex]\text{}y=f\left(x\right)[/latex], [latex]P=W\left(d\right)\text{}[/latex], and so on.
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