This assignment will assess your skills and knowledge on demonstrating algebraic operations, composition operations on functions and inverse functions, each with specific real-world applications. Algebraic operations are used in economics and physics, composition operations in computer science and physics, and inverse functions in cryptography and engineering. Understanding these concepts and their applications, answer the following questions clearly, showing step-wise solutions and providing reasons wherever necessary.

You are required to complete all the 4 tasks in this assignment. When you are instructed to make a graph in this assignment, please use

GeoGebra graphing tool

.

**Task 1.** Before working on this task 1, please read the following readings:

Read section 1.5 Function Arithmetic pages 76-92 of the following textbook to help you understand the concepts better and in easier way.

Stitz, C., & Zeager, J. (2013).

College algebra.

Stitz Zeager Open Source Mathematics.

https://stitz-zeager.com/szca07042013.pdf

- Read the following webpage to help you easily understand the operations of functions.

Mathisfun. (n.d.). Operations with functions.

https://www.mathsisfun.com/sets/functions-operations.html

If f: R→ Rand g: R→ R defined by f(x) = 2x+1 and g(x) = 3x+1.

(i) Find the functional values to the algebraic operations , (fg)(x), fog(x), and gof(x) by explaining the way of performing operations. Show all steps of calculations.

(ii) Are fg, fog, and gof equal? Please explain your answer.

(iii) Find the domain and ranges of each of the 4 operations. Explain the procedure of finding the domain and range for each of above 4 operations.

**Task 2.** In a cutting-edge greenhouse, an advanced climate digital control system is installed based on a DC inverter that maintains the desired temperature in the green house. The system’s temperature control function is described by

where T represents the temperature inside the greenhouse in degrees Celsius, and C represents the desired control setting on the climate control system of DC inverter.

Based on that information, please provide answers to the following questions:

(i) Using the concept of the inverse function, determine the control setting (C) as a function of greenhouse temperature (T). Please outline all the steps clearly.

(ii)Discuss any practical limitations or considerations that may impact the functionality of the inverse function within the context of the greenhouse climate control system.

**Task 3.**Before working on this task 3, please read the following readings:

- Read section 1.7 Transformations pages 120-139 of the following textbook will help you understand even and odd functions better.

Stitz, C., & Zeager, J. (2013). College algebra. Stitz Zeager Open Source Mathematics. https://stitz-zeager.com/szca07042013.pdf

If , here are some transformations , , , of

Please answer the following questions:

(i) Draw the graphs of , , , , of

ii) Explain the above 4 transformations of graphically.

iii) Write the observation on the domain and ranges of these 4 transformations of functions.

**Task 4.**Before working on this task 4, please read and watch the following resources:

- Reading section 1.6 Graphs of Function pages 95-99 of the following textbook may help you understanding the concepts better and in easier way.

Stitz, C., & Zeager, J. (2013). College algebra. Stitz Zeager Open Source Mathematics. https://stitz-zeager.com/szca07042013.pdfFor better understanding of the concept watch the video:Imagine Alex is examining a mechanical system and its displacement function. He has come across with the displacement function as . Please assist Alex in analysing the nature of the function.On the basis of the above scenario, answer the following questions:(a) What do you know about even function? According to you, how can Alex determine if the given displacement function, displays an even function mathematically? (b) Explain graphically in detail how the graph below possesses even symmetry or odd symmetry or neither.Submission Settings: You may use a word document that addresses the questions mentioned above. The word document should be double-spaced in Times New Roman font, which is no greater than 12 points in size. Read the rubric on how you are going to be graded on this assignment. Use APA citations and references if you use ideas from the readings or other sources. For assistance with APA formatting, view the Learning Resource Center: Academic Writing. Use high-quality, credible, relevant sources to develop ideas that are appropriate for the discipline and genre of writing.