GPS Alignment: Instructor Page
Week 7: Algebra  Functions
Composite Functions (Printable PDF)
Let w(x) represent the result of combining f(x) and g(x) into the composition function g(f(x)).
Let f(x) = 3x  7 and g(x) = 2x + 4.
The goal will be to find the function equation for w(x).
Start by inputting an xvalue into the f equation.
Take the output, or f(x), and use it as input into the g equation.
The resulting output is the output of the w equation, or g(f).
For example, let's start with x = 3.
Then f = 3(3)  7 = 2. Now put f(x) = 2 into g (note f(x) replaces x in this equation) and get = 2(2) + 4 = 0.
This means that w(x) will contain (3,0), where 3 is the original input and 0 is the final output.
Use a spreadsheet and generate several ordered pairs until you see a pattern that will help you write the function equation for w(x).
GPS 
As seen in problem exploration 
M6A2. Students will consider relationships between varying quantities.
a. Analyze and describe patterns arising from mathematical rules, tables, and graphs.

The student can use a spreadsheet to examine the relationship between the original functions and the new function. 
M7N1. Students will understand the meaning of positive and negative rational numbers and use them in computation.

This investigation involves examining functions and composite functions that contain both positive and negative numbers. 
M7A1. Students will represent and evaluate quantities using algebraic expressions.
b. Simplify and evaluate algebraic expressions, using commutative, associative, and distributive properties as appropriate.

The student will use and evaluate algebraic expressions, while creating the composite function.
The commutative and distributive properties can be applied when the student creates the composite function. 
M7A3. Students will understand relationships between two variables.
a. Plot points on a coordinate plane
b. Represent, describe and analyze relations from tables, graphs, and formulas.

The students can graph the points of the original functions and the composite function looking for a pattern. The process of writing the composite function requires that the student has an understanding of functions and relations and is able to analyze a function based on values in a table and graph.

M8A1. Students will use algebra to represent, analyze, and solve problems.
b. Simplify and evaluate algebraic expressions.

This investigation requires the student to combine the two algebraic expressions, simplify them and evaluate the composite function. The final step in the investigation involves writing the equation that represents the composite function. 
