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MATH 5a EXTRA PRACTICE FOR EXAM 2 1. Solve the following inequalities. Write your answers in interval notation. 3 2 a. x − 4x − 5x > 0 x2 − 2 b. ≤0 x−7 2. A graph is defined by the equation x2 + 2xy − y 2 = 4. Find the x- and y-intercepts (if any). 3. The following table shows the Super Bowl winners for the years 2002–2010. Is the year a function of the Super Bowl winner? Briefly justify your answer. Be specific! Year Super Bowl Winner 2002 New England 2003 Tampa Bay 2004 New England 2005 New England 2006 Pittsburgh 2007 Indianapolis 2008 New York Giants 2009 Pittsburgh 2010 New Orleans 4. Find the equation of the vertical line passing through the point (−2, 4). 5. Find the equation of the line that is perpendicular to the line 6x − 2y = 1 and passes through the point (2, −1). Graph both lines on the axes below. 6. The graph of a function f (x) is shown below. Use it to answer the following questions. f (x) (a) For what value(s) of x, if any, is f (x) = 0? (b) For what value(s) of x, if any, is f (x) = −4? (c) Find the domain of f (x). Write your answer in interval notation. (d) Find the range of f (x). Write your answer in interval notation. (e) On what interval(s) is the graph of f (x) increasing? Write your answer in interval notation. (f) On what interval(s) is the graph of f (x) positive? Write your answer in interval notation. (g) Find the average rate of change of f (x) from x = −3 to x = −1. (h) On the axes below, sketch the graph of y = − 21 f (x − 2) + 1. 7. Let f (x) = 1 f (x + h) − f (x) . Find and simplify as much possible. 3x − 2 h 8. Find the domain of the following functions. In part (a) write your answer using interval notation. In part (b) you may use any notation for your answer, as long as it is clear. √ x+2 (b) f (x) = 3 (a) f (x) = 3 − 17x x − 5x 9. Let f (x) = 4x − 1 . x + 3x + 1 2 (a) Find the domain of f (x). (b) For what value(s) of x is f (x) = 0? x + 4y = 1 1 if x < −1 x2 . 10. Sketch the graph of the following function on the axes below: f (x) = 3 x , if −1 ≤ x ≤ 1 √ x, if x > 1 y = 4x + 5 11. A study at an army training base found that the speed y at which a soldier can run when he is carrying equipment on his back is related to the weight x of the equipment by a linear equation. For example, a soldier carrying 20 pounds of equipment can run at a speed of 7 miles per, hour while a soldier carrying 40 pounds of equipment can run at a speed of 2 miles per hour. (a) Find the linear equation relating x and y. (b) What is the physical interpretation of the slope in this problem? Be very specific. (c) What is the x-intercept of the line you just found? What is its physical interpretation in this problem? 1 if x < −1 x+2 12. Sketch the graph of the following function on the axes below: f (x) = 1 . |x − 1|, if −1 ≤ x ≤ 2 2 4 − x, if x > 2 13. Determine whether each of the following statements is true or false. If a statement is true, briefly explain why. If a statement is false, else give an example (in the form of a formula or a picture) that shows that it’s false. (a) The graph of a function can have two y-intercepts. (b) Suppose that f (x) is a function whose graph is not a straight line. If the average rate of change of a function f (x) between x = a and x = b is positive, then f (x) must be increasing on the interval (a, b).