Task 1: If f: R Rand g: R R defined byf(x) = 2x+1andg(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 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 T (C ) 1}%'281%2 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. (i)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 f(x) = v/z, here are some transformations /z + 6, /= 6. \\/50z v/x/50 of /z Please answer the following questions: (i) Draw the graphs of f(z) = /. /z + 6./ 6. /501 \\/W of /x ii) Explain the above 4 transformations of \\/1_ graphically. iii) Write the observation on the domain and ranges of these 4 transformations of functions. Imagine Alex is examining a mechanical system and its displacement function. He has come across with the displacement function as glt) = T ):\"'} . 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, g(#) = % displays an even function mathematically? ail how the graph g(t) = i ):\": below possesses even symmetry or odd symmetry or neither. T 5 4 3 2 1 - 7 6 5 4 a3 2 0 1@:F 9 (b) Explain graphically in de