Composite Functions
Note: Composite Functions
Composition of one function on another.
Given f: x y, g: y z; then, g
f: x z is defined by g
f(x) = g(f(x)) for all x.
(Read “g composed with f” or “g circle of f”, or “g’s of f”) f(x)=x+1 g(y)=2x+1 x y z
0 1
1
2
2
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Example: Given f(x) = x + 3, g(x) = x 2 , find
1.
f(g(2))
2.
g(f(2)
3.
f(g(-2)
4.
g(f(x))
3
5
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Composite Functions - f(g(x)): Lab work
Objectives: Investigate and discover the concept of composite functions/ domain and range.
Materials: bread slices (whole wheat and white), cookie cutters, and cheese-it spread.
Procedure:
1. Split the class into 4 groups: A, B, C, D. Each group should select a recorder, a runner, a cutter, a spreader, a timer, and a leader. The leader ensures that each task gets done correctly..
2. Each group sends it’s runner to the supply table to pick up 4 slices of both types of bread, 2 cookie cutters, and a can of cheese-it spread.
Each group will fill in the lab sheet below. Timer - You have 15 minutes to complete the task below.
Group Members: Leader –
Recorder –
Runner –
Cutter –
Timer –
Spreader –
Activity 1 - Function f (shape formation) – The Cutter cuts 2 slices each of whole wheat and white bread into shapes using the cookie cutter. Discuss with the group what the input (domain) and the output (range) of the function of this shape formation activity should be and the recorder writes down in the table on the back.
Activity 2 - Function g (cheese-it spreading) - The spreader spreads a bit of cheese-it on the uncut slices of bread. Consider the domain and the range of the function of spreading cheese-it. The
recorder writes down in the table on the back.
Activity 3. – Function g (f (bread)) - The spreader spread cheese-it on the cut pieces of bread.
Discuss and record the domain and range in the table on the back.
Activity 4. – Function f (g (bread))– The cutter presses the cookies cutter and shapes the slices of bread with cheese-it. Discuss and record the domain and range in the table on the back.
CHALLENGE: Activity 5. – As a group consider a function, m (cheese-it spreading), where it’s domain is
{heart shaped bread}. What would be its domain and range?
What is your reasoning?
Function Domain Range f: shape Formation g: cheese-it Spreading g(f(bread)) f(g(bread))
CHALLENGE: Activity 5
