# 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

process.

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.

**Model #1
Function Chains**

**Model #2
Change of Variable**

**Model #3
Modified T-table**

**Model #4?**

**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.