Blog Post #4

Function

The purpose of this lesson is to teach students what a function is, how it can be shown as and what are and what aren't functions. I give a real life example that will clearly demonstrate what a function is and how it can be represented as using words, data in a table, points on a graph or a formula. 

In mathematics, a function is a rule that takes certain numbers as inputs and assigns to each input number exactly one output number.

·      The output is a function of the input à output = f (input)

·      If the value of the first quantity determines exactly one value of the second quantity, then the second quantity is a function of the first.

·      A function can be described as using words, data in a table, points on a graph, or a formula.

Example (from Textbook)

It is a surprising biological fact that most crickets chirp at a rate that increases as the temperature increases. We can estimate the temperature in degrees Fahrenheit by counting the number of times a snowy tree cricket chirps in 15 seconds and adding 40. For instance, if we count 20 chirps in 15 seconds, then a good estimate of the temperature is 20+40=60 degrees Fahrenheit.

The rule used to find the temperature T from the chirp rate R is an example of a function. The input is chirp rate and the output is temperature. This function can be described using words, a table, a graph and a formula as shown below:

Using Words

To estimate the temperature, we count the number of chirps in fifteen seconds and add forty. Alternatively, we can count R chirps per minute, divide R by four and add forty. This is because there are one-fourth as many chirps in fifteen seconds as there are in sixty seconds. For instance, 80 chirps per minute works out to ¼ x 80 = 20 chirps every 15 seconds, giving an estimated temperature of 20+40 = 60 degrees Fahrenheit.

Using a Table

Table 1.1 on textbook page 3 shows the R chirp rate (chirps per minute) on the left column and on the right column is T temperature. The left column R chirp rate begins with the value 20 and increases in increments of 20 up to 160 while the T temperature column on the right increases by 5 degrees Fahrenheit starting at 45 and ending at 80. Each time the R chirp rate increases by 20, the T temperature increases by 5 degrees Fahrenheit.

Using a Graph

Graph 1.1 on textbook page 3 is a graph where the y-axis is the T temperature starting at 10 degrees Fahrenheit increasing by increments of 10 and ending at 100 degrees Fahrenheit. The x-axis is the R chirp rate that begins at 40 and increases by increments of 40 ending at 160. For example, the pair of values R = 80, T = 60 is plotted as a point on the graph. That point is 80 units along the horizontal axis and 60 units up the vertical axis. The precision of that point plotted is shown by its coordinates written as (80, 60).

Using a Formula

Dividing the chirp rate by four and adding forty gives the estimated temperature so the formula to find the estimated temperature would be:

                        T = ¼ R + 40

Mathematical Model

When the output is dependent on the input then we call a function a mathematical model. Mathematical models are functions used to describe actual situations such as estimating the temperature by using the function for the snowy tree cricket. So for example, as the chirp rate increases the estimated temperature will increase, and as the chirp rate decreases the estimated temperature will decrease. Therefore estimating the temperature is dependent on the chirp rate.

Function Notation

Function notation is writing out a function in relation to the independent and dependent variables. In other words, because the output is a function of the input the basic function notation for this statement would be:

                        Output = f (input)   or   Dependent = f (independent)

When is a relationship not a function?

It is possible for two quantities to be related but at the same time not be a function of one another.

·      A relationship is not a function when there is more than one output per input.

·      A relationship is not a function when the values are graphed, the line created will not pass the vertical line test. So, when you have a graph, draw a single vertical line and if the line of the graph only passes through your vertical line once then it is a function, if it passes through your vertical line more than once it is not a function. For example a circle graphed on a Cartesian coordinate plane will not pass the vertical line test therefore it is not a function.