Ejournal 3: “A Function of Functions”

A function is a relationship or expression involving one or more variables.  It is a relation wherein a set of inputs to a set of possible outputs where each input is related to exactly one output. In other words a function is a one-to-one or a many-to-one function (shown below).

An easy way to deduce if a graph is a function is by the vertical line test. This test is simply that if a vertical line at any point of the graph intersects the equation at the two points it is not a function. Some examples of functions are y = 3 and y =3x + 5.

These functions can further be transformed into other functions which look similar to the original function but are for example shifted to the left/right, up/down, reflected and/or horizontally/vertically stretched/shrink according to the change in function (as seen in the image below).

g(x) = f3

h(x) = f7

j(x) = f5

k(x) = f1

p(x) = f4

r(x) = f3 & f8

s(x) = f9

t(x) = f6

v(x) = f2

I could match the function to the line based on the characteristic of the equation. Equations like g(x) & r(x) were straight horizontal lines as it was y = a. I could deduce what j(x), k(x) & s(x) was based on the domain given. Domain is a restriction on the range of possible x values a function can have. The remaining functions, h(x), p(x), t(x) and v(x) are the formulas for semi-circles so I could match them accordingly based on the negative sign and the values inside the function which I will explain shortly.

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I used the Desmos application to graph this. I changed some parts of the original functions to fit the range desired, and also based it on the structure of a real bicycle including the pedals, gear and spokes I graphed. The functions depicted in the photo above include linear, semi-circular and square root functions (which will be explained shortly).

Linear Equation:

y = mx + b; where m is the slope and b is the y-intercept

Semi-circular equation:

where b is the height of the lowest point in the semi-circle from the y-axis, r is the radius of the semi-circle, and a is the x-coordinate of the highest point of the semi-circle

Square root equation:

The image below describes some of the changes in the shape of the function based on the changes in the variables.

Real life application

1. Forensic scientists use functions to help determine the height of a person based on the length of the femur. The function h(f) = 2.47 + 54.10, ±3.72cm. Where 𝑓 is the length of the femur. I find this example really exciting because I have watched numerous forensic TV shows which use this technique but didn’t exactly know it was an application of functions.
2. To determine the position of an object the distance an object travels as a function of time is given by 𝑠(𝑡)=12𝑎𝑡2+𝑣0𝑡+𝑦0𝑎, where a is the acceleration due to gravity (−9.81232m/s^2), 𝑣0 is the initial velocity and 𝑦0 is the initial height. This formula is used by many people who take physics.
3. In determining the price of objects such as the price of a car, it depends on the cost of many variables or raw materials: metals, plastics, glass, salary, location of the company, taxes, quality of manufacturing, etc. There are functions which include these variables to help determine the selling price. Similarly, the climate of any region is a function of temperature and air pressure, density, ocean-atmosphere interaction, CO2 presence, altitude, etc.

In conclusion, we can see that functions like other areas of math are often used in our daily lives, whether we realize it or not. I hope that this blog post has sparked your curiosity in thinking about how other math topics you have learnt so far can be relevant in real life not just for learning at school to pass a test.

IB Learner Profile

1. The learner profile that resonates with me the most during the production of this ejournal is reflective as at first I lost all the functions I used when I quit the application. Thus, I had to reflect on what I had previously done and ponder on how to further improve the bicycle even though I was really disappointed that I had to restart my work again.
2. Another learner profile that I can relate to is thinkers, as some of these functions like the one for producing a semi-circle were new to me and I had to figure out how to manipulate the values into giving me the graph I wanted. I had to persevere in testing different values of variables to see the effect it had on the graph, and from that I was able to implement my new found knowledge.
3. The last learner portfolio is inquirers as during the process of making the bicycle, I wanted to make a sideways parabola for the gear. I had to research how to make one even though it was not required, but after making it I realized it was not a function as a vertical line would pass the graph twice. In the end for making the gear, I used two different functions to replace the one equation that was not a function

Sources:

https://mathinsight.org/definition/function

https://www.quora.com/What-are-the-examples-of-an-application-of-a-function-in-real-life

https://www.quora.com/What-are-the-applications-of-functions-in-real-life