** Step-by-step solved example for the Composition of functions**

**Here are some Discussion assignment clarifications, with a solved example, for this week’s topic:**

Since most of the people have questions about the composition of functions, we have to work together on this assignment. Please review **section 3.4** for the composition of functions’ definitions and examples, as this subject is challenging, but interesting, and useful, and requires some very good analytical and attention to detail skills.

**Note 1**:

Somewhere, in the fine lines, I stated that the temperature conversion is just a starting point for our composition of functions’ analysis. **The only topic that we discuss here is about the process of composition of functions** which is not an easy subject. Please note that a conversion formula (C to F or F to C) is considered the function for our practice exercise.

**Please see below my solved example, with detailed explanations and comments. I will choose the month of October, where the average temperature in my case is m = 63, (M or m=month). **

**Find:** F(Fahrenheit) = F(M) = F(Oct) = F(63) = 63

C(Celsius) = C(M) = C(Oct) = C(63) = 5/9(M-32) = 5/9(63-32) = 5/9 x 31= 155/9 = 17.2

**1.** Calculate (C ᵒ F)(M) for the month of your choice (October), and show formula and steps:

(C ᵒ F)(M) = (C ᵒ F)(63) = C(F(63)) = C(63) = 155/9 (it is OK to repeat the steps)

**2.** Discuss the meaning of the function (C ∘ F)(M) = C(F(M))

A “**Function Composition**” is applying one function to the results of another function. In this operation the function C is applied to the result of applying the function F to M. (Please do not limit the discussion to the topic of temperature conversion only as we can have any letters/variables for our practice exercises).

**Note 2**:

Since we can have any letters/variables for our functions it is not important what a temperature conversion is but how do we solve a composition of functions, which can become complicated fast. In other words, how do we de-compose (break up) a function into a composition of other functions? How do we make things simpler in order to find a solution?

**3.** How does the composition of functions in parts 1 and 2 compare to (F ∘ C)(M)? Are they the same? or are they inverse?

**a).** Composition of Celsius and Fahrenheit (result in Celsius):

(C ᵒ F)(M) = (C ᵒ F)(63) = C(F(63)) = C(63) = 5/9(**63**-32) = 5/9×31 = **155/9**

**b).** Composition of Fahrenheit and Celsius (result in Fahrenheit):

(F ᵒ C)(M) = (F ᵒ C)(63) = F(C(63)) = F(155/9) = 9/5C+32 = 9/5×155/9+32 = 31+32 = **63**

**As shown above the two functions are inverse.**

**Note 3**:

We use the Celsius conversion formula to solve for variable F (Fahrenheit):

C = 5/9(F-32) Celsius (from Fahrenheit)

F = 9/5C + 32 Fahrenheit (from Celsius)

For C = 155/9 (easier to simplify if we keep the results as a fraction) we have:

F = 9/5×155/9 +32 = 31+32 = 63

**Note 4: Final Observations**

In a business setting you have a maximum of 3 or 4 minutes to present your work. This is why your comments must be concise and precise. Also, you have to be able to say a lot with very little. In terms of analytical and attention to detail skills, I think this assignment is one of the best, and an eye opener, since Algebra is the second most unforgiving type of mathematics (second only to Statistics).

I end my comment with a Composition of functions’ **Summary**:

- A “Function Composition” is applying one function to the results of another.
- (g º f)(x) = g(f(x)), first apply f( ), then apply g( )
- Some functions can be de-composed into two (or more) simpler functions.==============================================================================================================================================================================QUESTION
- Consider the following two functions:
*F*(*M*): The average temperature (*F*) in Fahrenheit during month (*M*) of the year.

**Month (***M*)*F*(*M*)January 42 February 43 March 52 April 64 May 72 June 83 July 86 August 84 September 73 October 65 November 55 December 44 *C*(*F*): The conversion formula to calculate the temperature in Celsius (*C*) based on the temperature in Fahrenheit (*F*).

C(F)=59(F−32) $C\left(F\right)=\frac{5}{9}\left(F\u201332\right)$Your task for this discussion is as follows:

- Calculate (
*C*∘*F*) for the month of your choice. - Discuss the meaning of the function (
*C ∘ F*)(*M*). - Now think about the composition, (
*F ∘ C*). Does this composition make sense? If so, how does it compare to (*C ∘ F*)? If not, why not?