Composition of Functions
A typical Algebra class only discusses composition of functions algebraically. This dynamic worksheet demonstrates compositions as translations of graphs of functions.
Materials: Computer Access for each student (Students access the following website- http://www.geogebra.org/en/upload/files/UC_MAT%202009/Jillian%20Maher/Algebra%20II/composition_1_.html )
Attached worksheet copied for each student
Length: Thirty to Forty Minutes
Procedure:
Students begin by exploring the sliders on the dynamic worksheet using section I of the hard copy worksheet. They will set F(x) and G(x) to be any functions they choose and then algebraically calculate the compositions F [G(x)] and G [F(x)]. The answers the student produces should be checked using the checkboxes for the composition graphs. Notice that the students are not given the algebraic answer to the composition – only a graph. They must be able to identify if the graph is the same as their equation.
Any observations they notice comparing the function G(x) to the composition graphs should be written in section II of the hard copy worksheet. Students should look for shifts in the graph as well as stretching and shrinking.
While the first two sections only familiarize the students with the worksheet, the final section should help students better understand the graphical interpretation of these compositions. Students will “fix” certain functions and watch the changes to G(x) and the composition functions. Most of the “fixed” functions are F(x). This linear fixing will help students to identify translations. They should notice that when F(x) is the inside function, there is a horizontal shift from G(x) to the composition. Likewise, if the fixed function is the outside function, there is a vertical shift from G(x) to the composition.
Even bigger than identifying the type of shift involved, students should notice the value of the shift. When there is a horizontal shift, students should recognize it is the opposite value of the function. For example, when F(x) = x + 2 is the inside function, the graph of G(x) shifts two units to the left. However, when F(x) = x – 3 is the inside function, the graph of G(x) shifts three units to the right. While the horizontal shift has an opposite valued shift, the vertical shift is a direct valued shift.
Composition of Functions NAME______
I. Guess Work
Move your sliders to form three different functions for each F(x) and G(x). Use the table below to write out your F(x), G(x), and compute each of the compositions F(G(x)) and G(F(x)). Be sure to simplify the functions. Also write what you think the composition graphs will look like, giving any details specific to that style of graph (for example, y-int, vertex, roots, etc).
F(x) = / G(x) = / F( G(x) ) = / G( F(x) ) = / What will the graph be? – give details if possible.1
2
3
II. Check yourself
Now use the checkboxes to review the graphs of your functions above. Were you right? How close were you? How do the graphs of F(G(x)) or G(F(x)) compare to the graph of G(x)? Write down your observations in the table below – be specific!
Observations1
2
3
III. Extend it!
Use the table below to make one of the functions F(x) or G(x) unchanged, and change the alternate function using its sliders. Write your observations comparing the graphs of the composition functions to that of G(x). Make sure the composition graphs are checked to see them. HINT: If you are having a hard time finding out what is happening when F(x) is fixed, set a = 0 for G(x) and only change the other two sliders.
Fixed Function / Observations of F(G(x)) / Observations of F(G(x))1 / F(x) = x + 2
(HINT: set m =1
and b = 2)
2 / F(x) = x – 3
3 / F(x) = x + 1
4 / F(x) = x - 5
5 / F(x) = -x
6 / G(x) = x2
7 / G(x) = 2x2 + 4x + 2
What did you find overall? What happens when F(x) is the inside function? What happens when F(x) is the outside function? On the back of this page, write a paragraph about your findings.