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Math Problems 1. Which of the following are functions? Explain your reasoning for a, b, and c.a. f(x)=1 if x>1f(x)=0 otherwiseb. f(x)=2 if x>0f(x)=-2 if x<0f(x)=2 or -2 if x=0c. f(x)=5/x2. Suppose you have a lemonade stand, and when you charge \$1 per cup of lemonade you sell 60 cups. But when you raise your price to \$2 you only sell 30 cups. Write an equation for the number of cups you sell as a function of the price you charge. Denote "C" for number of cups, and "P" for the price you charge. Assume the function is linear.3. Take a look at the table below and write out an equation for f(x).x -2 -1 0 1 2f(x)-1 1 3 5 74. For each of the relationships below, explain whether you think it is best described by a linear function or a non-linear function. Explain your reasoning throughly.a. How fast it takes you to get to work as a function of how fast you drive.b. Probability of getting into a car accident as a function of how fast you drive.c. Your height as a function of age (from age 0 to 100) 5. find the area and circumference of a circle whose diameter measures 12 centimeters6. Find the area of a cylinder whose radius is 2 inches and whose height is three inches7. The maintenance bill for a shopping center containing 180,000 square fee is \$45,000. What is the bill for a store in the center that is 4,800 square feet?8. A new car costs 125% of what it cost 4 years ago. Determine the price 4 years ago if the car now sells for \$32,500. 9. Use the Graph of f(x)=x^2 to sketch the graph of the function by hand. verify with a graphing utility.g(x)=(x+1)^2-3Q. 10. Determine whether the function has an inverse function. If it does, find its inverse functionf(x)=36+x^2, x<=0 12. Find the solution for the following factors. Write your answer in set notation.(3x+5)(4X-3)=0 13. Determine the open intervals over which the function is increasing, decreasing, or constant.f(x) = square root x squared - 9 14.Consider the polynomial function f(x)=4x^4-13x^2+9a. What is the maximum number of real roots you would expect and why? b. Find the real roots of the polynomial function. Using factoring by substitution, or substitution and the quadratic formula to find the real roots. 15. Factor completely:(a) x^2 - 12x + 27 =(b) x^3 - 16x = 16. Solve & simplify completely:(a) x^2 = 49(b) (y - 3)^2 = 1617. Solve & simplify completely:(a) p^2 = 48(b) 7x^2 = 4218. Solve & simplify completely:(a) 2z^2 + 3z = 0(b) (x - 3)^2 = 719. Solve by completing the square:x^2 - 6x + 8 =020. You borrow \$100 loan P for two years and the interest is compounded annually. Your payoff A, is \$121. Find the APR. Explain every step, your reasoning and what you are doing. Do NOT just give formula and answer, if you want credit.21. The width of a rectangle is 2 feet shorter than its length. If the area is 16 square feet, what are its dimensions, both, length and width. Again: Explain every step, your reasoning and what you are doing. AGAIN: Do NOT just give formulas and answer, do not use calculators - leave roots, but simplify completely.22. Solve each system by the substitution method. Determinewhether the equations are independent, dependent, or incon-sistent.3(y - 1) = 2(x - 3) 3>y- 2x = -3 23. An apartment complex has 100 apartments. When the rent is \$300 all of the apartments are rented. However, for every \$10 increase in the rent above \$300, two additional apartments go vacant. The revenue equation for this apartment complex is R(x) = 30,000 + 200x – 20x2, where “X” is the number of \$10 increases in rent. What rent will result in the greatest possible revenue? What is the greatest possible revenue? Graph this function and identify the axis of symmetry, vertex, x-intercepts, and y-intercept.24. Solve the following system of simultaneous equations, where x, y, z 2 C :    x + y + z = 0    x^2 + y^2 + z^2 = 12   x^3 + y^3 + z^3 = 27 25. Find, with explanation, all integers x, y with x(y + 1)2 = 243y 26. Simplify the expression and show your work:    (-54a^21b^23/-27a^19b^24)^3 27. Answer in complete sentences:a) Explain what is the y-intercept and which letter represents it in the equation y = mx +b.b) If a and b are positive numbers, what is all the information we can deduce about the relationship between the lines: L1: y = - ax + b and L2: y = (1/a)x + b? 28. Let f(x)=5/x and g(x) = 3 – 2x(a) Find f o g (b) What is the domain of the function found in (a)?(c) Calculate (f o g)(1/4)For the functions f and g given above,(a) Find f o f and simplify as much as possible.(b) Find g o g and simplify as much as possible. 29. This is a problem involving (fog) and includes a square root. I do not have the ability on my computer to properly show the square root sign, so I will show it like: "square root ( )".....with the items under the square root sign in the parenthesis.Here's the problem:Suppose f(x) and g(x) are as follows:   f(x)=1/x   g(x)=square root(x+2)a) What is (fog)(x)=?b) What is the domain of (fog)(x)? 30. Here is a rational function    f(x)=9x/1+3xa) What is the domain?B) When x is approaching infinity, y approches what?c) What is the equation of the vertical asymptote?d) Is there a vertical asymptote? If so what is the equation? (Looks like my instructor is repeating himself here. I wonder if he actually means horizontal?)e) What are the x and y intercepts?f) Sketch a graph of the function.        