Composition of Functions
Part 1: An Example
1. There are 3 scales used in measuring temperature –
Fahrenheit, Celsius and Kelvin. One can
convert Fahrenheit reading to Celsius using , where t represents temperature
in degrees Fahrenheit. Likewise, Celsius readings can be converted to Kelvin using
k(t) = t + 273 , where t represents temperature in degrees Celsius.
a. I want to convert 100° F to the Kelvin scale. Make the
conversion and explain your
b. Convert − 5 ° F to Kelvin
c. Convert 115° F to Kelvin
d. Generalize this process by creating a function that
will take measurements in degrees
Fahrenheit and convert them into degrees Kelvin, and explain how you did that.
Change of Variable
Part 2: Some Big Ideas about Function Composition
1. In mathematics, combining simple functions with
"composition" to form more complicated
functions is somewhat similar to the way that in chemistry elements can be combined
through chemical bonds to form compounds.
2. Function composition combines a pair of functions f and
g into a single "composite"
function written f o g . The composite function operates in a chain where the output of
function g is used as the input of function f.
3. Sometimes the order of the composition of two functions
makes a difference, while at other
times it doesn't. That is, for some functions f and g, f o g = g o f while for other functions f
and g, f o g ≠ g o f
4. There are three common operations by which two
functions f and g can be combined to
form another function h: addition ( f + g) , multiplication ( f * g) , and composition f o g .
The three are quite different, and function composition needs to be clearly distinguished
from the other two.
5. Given a graph of a function f, graphs of f (x + 2) and
f (x) + 2 represent translations of this
graph, while graphs of f (2x) and 2 f (x) represent expansions or contractions of this graph.
These transformations of graphs can be expressed in terms of the composition of f.
6. Iteration (repeating a process over and over) can be
expressed in terms of the composition
of a function with itself.