Relations and FunctionsIn this discussion, you will be assigned two equations with which you will then do a variety of math work having to do with mathematical functions. Read the following instructions in order and view theexample to complete this discussion:  MAT222.W5.DiscussionExample.pdfProblems are found circled in the following attachment: Week 5 Discussion instructions.pdf  There are many ways to go about solving math problems. For this assignment you will be required to do some work that will not be included in the discussion. First, you need to graph your functions so you can clearly describe the graphs in your discussion. Your graph itself is not required in your post, although the discussion of the graph is required. Make sure you have at least five points for each equation to graph. Show all math work for finding the points.Specifically mention any key points on the graphs, including intercepts, vertex, or start/end points.  (Points with decimal values need not be listed, as they might be found in a square root function. Stick to integer value points.)Discuss the general shape and location of each of your graphs.State the domain and range for each of your equations. Write them in interval notation.State whether each of the equations is a function or not giving your reasons for the answer.Select one of your graphs and assume it has been shifted three units upward and four units to the left. Discuss how this transformation affects the equation by rewriting the equation to incorporate those numbers.Incorporate the following five math vocabulary words into your discussion. Use bold font to emphasize the words in your writing (Do not write definitions for the words; use them appropriately in sentences describing the thought behind your math work.):FunctionRelationVertical Line testTransformationYour initial post should be at least 250 words in length.

1.  In your chemistry courses, you will encounter the term ‘pH’, which measures acidity of a solution.  Neutral is 7, above 7 is labeled Basic, and below 7 is Acidic. We define:   pH = - log [H+] (which we read ‘Concentration of hydrogen ions in moles per liter of solution’). A mole is approximately 24! ions, though you don’t need to know that for this problem.  But you do need to know these facts:  I: The pH of human blood is 7.4;  II: The pH of Coca Cola is 2.5; III: Distilled water has pH 7;  IV: When we write ‘log’ we mean base-10. IV: One ml (milliliter) is 10-3 liters. (i)  The human body has a blood volume of 5 liters.  So how many moles of H+ are in your bloodstream?  Write the answer in scientific notation. (First figure out how many moles per liter!) (ii)  Now suppose we make a vampire cocktail punch by mixing blood of 199 humans with 5 liters of Coke. Find (a) the total number of moles of H+, and (b) the total volume of punch being made for the party.  Then (c) compute the pH of the punch. (iii)  Next, let’s deal just with Coke and water. We want to make a very flat punch for a party attended by accountants. If we mix x liters of water with 1 ml of Coke, what is the volume of punch, and how many moles of H+ ion does the punch contain? (iv)  Use (iii) to write a formula for the pH of the punch (in terms of x). (v)  Now use your work in (iii) to determine how many liters of water you will need to add to 1 ml of Coke, to increase the pH of the punch by one unit above that of straight Coke. (vi)  Express the answer to (v) in a complete declarative English sentence. 2.   Suppose we have a population that increases according to the formula P(t) = P0 ekt.  Our goal is to fill out the following table: Point t P(t) A 0 B 10 20 C 30 50 D 40 E 2000 (i)  Use points A and B to write two equations.  Then use them to solve for P0 and k.  Finally, write the population equation. (ii)  Use (i) to fill in the rest of the chart.  You may round each value to the nearest tenth. (iii)  Now use your formula to determine the doubling time of this population. (iv)  Explain how, without using an equation, you can now calculate the t values for which P(t) equals 1000. And how about 250?  Do this by writing a clear brief essay, with complete English sentences. (v)  Now on graph paper, draw a clear graph of this exponential function. Be sure to label your axes, and to label, with their coordinate, each of points A through E. 3.  (i) Solve exactly:  22x-1 = 11x+1.  (Hint: It is probably easiest to start by taking logs of each side.) (ii) (a) Consider the function f(x) = ln (x - 1). Find its domain.   (b) Now find the composition f ( f(x) ). Again, figure out its domain. (It may help to draw one of those flow diagrams we use in class for compositions. Work backwards.   (c) Once more. Extend your work to find the domain of f(f(f(x))). 4.  Consider the following situation:  A circle of with center O(0,0), radius 10m, is inscribed in a square. The ray of angle 30O, in standard position, intersects the circle at point B, and continues to intersect the square at point C. Let A denote (2,0).  (i)  Sketch the figure indicated in the above description. (ii)  Find the exact coordinates of A, B, and C, and label them on your sketch. (iii)  Now suppose we have arbitrary acute angle Q  (in radians, instead of the 30O).  Again draw the sketch! (iv)  Again figure out the exact coordinates of A, B, and C and label them on your sketch. NB: You will use trig functions here! (v)  Now figure out the equation you would have to solve to find Q to make the area of ABCA exactly equal to the area of the sector.  HINT: This means area of sector is half area of triangle. (You cannot solve such an equation exactly – this is an example of a TRANSCENDENTAL equation, so the theorems of algebra do not apply.)

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