## Exploration 01 - Exploration of Function Graphs

I have created two different files to play with on this one.

First, one can play with sliders in Desmos via the following link: https://www.desmos.com/calculator/1zwd9jfu4n

You can also use the following GSP files (and change the functions listed to understand the impact:

First, one can play with sliders in Desmos via the following link: https://www.desmos.com/calculator/1zwd9jfu4n

You can also use the following GSP files (and change the functions listed to understand the impact:

functions.gsp | |

File Size: | 12 kb |

File Type: | gsp |

I think the most important thing to clarify with students is that the f(x), g(x), or h(x) symbols represent the OUTPUT of a function. The x does signify the input, but f(x) signifies what is returned from the actions of the function.

## f(x)+g(x)

While many resort to algebra and tables to represent adding up outputs of two functions or combining two linear functions into one linear function, I think it is important to depict what happens geometrically in the coordinate plane as well. We are simply adding the y line segments together at each x input. Please watch the video below to see.

I also think it might be interesting to have students depict and understand dynographs in this exploration:

http://www.dynamicgeometry.com/JavaSketchpad/Gallery/Trigonometry_and_Analytic_Geometry/Dynagraphs.html

I also think it might be interesting to have students depict and understand dynographs in this exploration:

http://www.dynamicgeometry.com/JavaSketchpad/Gallery/Trigonometry_and_Analytic_Geometry/Dynagraphs.html

## f(x)*g(x)

This one is a bit harder to see and could be done using some right triangles. We are multiplying the output, but when you multiply the linear functions, you realize that your new function is a degree higher and results in a parabola. The rate of change is not consistent anymore due to the varying magnitude of the outputs. I have created another video to review:

## f(x)/g(x)

This is a hard one for students to wrap their heads around as they need knowledge of polynomial division, roots, and rational functions to comprehend. The result is a rational function as you have two irreducible linear functions in the division. (Hey, technically we could create a field here if we wanted to jump to abstract algebra!). I will note, GSP does not demonstrate a vertical asymptote that should exist in my function at x=2, so it might not be a good tool to use. It is very important for students to understand that there can be no output or solutions at x = 2 as the h(x) would then be undefined.

I mentioned using Dynagraphs to demonstrate these functions and outputs better. Below, I have posted a sample of one I generated in my exploration.

I mentioned using Dynagraphs to demonstrate these functions and outputs better. Below, I have posted a sample of one I generated in my exploration.

## h(x)=f(g(x))

This is always a rough one for students to wrap their heads around often. Let's say we have the following linear functions:

f(x) = ax+b

g(x) = cx+d

We have to remind students here, that our INPUT is x, and our output is ax+b or cx+d here.

So, in the case of f(g(x)), the INPUT of function f will be the output of the function g. (The x of f(x) will be g(x) or the output, cx+d). So, our input in the function is now cx+d, or f(cx+d).

Now, we must find what our output of function f is, given our input:

f(g(x))=f(cx+d)=a(cx+d)+b

f(g(x))=(ac)x +(ad+b)

Because (ac) and (ad+b) are constants, you see we maintain the format of a linear function and our function remains linear in this case.

I think the best way for students to explore dealing with the output of functions is to play with some of the tools designed and compare the results to results they are finding using algebra. I think it is also important here to begin discussion of injective (1-1), surjective (onto), and bijective (1-1 and onto) functions during this lesson as well. What are each of the functions discussed thus far?

f(x) = ax+b

g(x) = cx+d

We have to remind students here, that our INPUT is x, and our output is ax+b or cx+d here.

So, in the case of f(g(x)), the INPUT of function f will be the output of the function g. (The x of f(x) will be g(x) or the output, cx+d). So, our input in the function is now cx+d, or f(cx+d).

Now, we must find what our output of function f is, given our input:

f(g(x))=f(cx+d)=a(cx+d)+b

f(g(x))=(ac)x +(ad+b)

Because (ac) and (ad+b) are constants, you see we maintain the format of a linear function and our function remains linear in this case.

I think the best way for students to explore dealing with the output of functions is to play with some of the tools designed and compare the results to results they are finding using algebra. I think it is also important here to begin discussion of injective (1-1), surjective (onto), and bijective (1-1 and onto) functions during this lesson as well. What are each of the functions discussed thus far?