Exam 1 Review Math 133 Study Resources The first exam is Wednesday, February 9 in class. We will have time in class on Monday to go over some of the material and questions in this review packet, but you are encouraged to come to office hours or the Math Assistance Center for extra help. Math Assistance Center (MAC) hours: – Sunday-Thursday, 6:00 - 9:00 pm Go over homework problems, old worksheets. Make sure you understand how to solve those problems on your own, without following along somebody else’s work, and come ask about any that you are unsure on using the above resources. The list below is not a comprehensive list, but just a primer to get you thinking about the topics that we have discussed through the past few weeks. The test may cover any and all material we have covered in chapters 3 and 4, and will likely have more real-world application questions than are included in this packet. Key Terms, Concepts, Ideas, Formulas, etc. A function is a relation where every has exactly one . In the statement “Temperature is a function of time of day”, the input of the function is and the output of the function is The set of all inputs to a function is the function is the A function is called . . The set of all outputs of a . if every input has a different output (alternately, if no output is repeated by different inputs). When finding the domain of a function, there are two major “problems” to watch out for: – When our function involves a fraction, we cannot . – When our function involves a square root, we cannot . The average rate of change of the function y = f (x) on the interval [x1 , x2 ] is given by the formula Average rate of change = A point at which a function switches from decreasing to increasing is called a The absolute maximum of a function is the function (if it exists). 1 point on the graph of the . A function is called even if f (x) = . A function is called odd if f (x) = . If f (x) is a function, then – f (x) + k is a – f (x + h) is a – af (x) is a – f (bx) is a A valid order for the transformations in the function af (b(x − h)) + k is 1. 2. 3. 4. If f (x) is a one-to-one function, then f −1 (x) is the of f (x), which switches the inputs and outputs of the original function. A linear function is a function with a rate of change. The graph of a linear function is a line. The slope-intercept form of a line is The point-slope form of a line is Two lines are parellel if they don’t meet. Two lines are parallel if the have the same and different Two lines are perpendicular if they meet at right angles. Two lines are perpendicular if they have slopes. 2 Exam 1 Review Math 133 Review Exercises 1. Determine if each of the following graphs corresponds to a function. If so, determine if that function is one-to-one. (a) (b) (c) 2. Determine the domain of the following functions. (a) f (x) = x−3 x2 − 4x − 12 (b) g(x) = √ x−6 3. The table below shows the relationship between car model and price. Using the information, is car model a function of price? Is price a function of car model? Sentra $20,000 Malibu $24,000 Corolla $21,000 Jetta $20,000 Forte $19,000 Elantra $21,000 Fusion $25,000 Charger $31,000 3 4. For each of the following graphs, (a) Determine Domain and Range (b) Determine intervals of increasing and decreasing (c) Find all local maxima and minima (d) Find the absolute maximum and minimum, if they exist. −6 −4 6 6 4 4 2 2 −2 2 4 −6 6 −4 −2 2 −2 −2 −4 −4 −6 −6 4 6 5. For each of the following functions, compute the average rate of change on the given interval. (a) f (x) = x2 + 3x − 2 on (b) g(x) = 4|x + 1| − 3 on the interval [−5, −1] the interval [−3, 3] 4 Exam 1 Review Math 133 6. Find f (g(x)) and g(f (x)) for each of the following pairs of functions. (b) f (x) = x2 + 2x, g(x) = 5x + 1 (a) f (x) = 3x + 2, g(x) = 5 − 6x 7. The function f (x) is given by the piecewise formula below, and the graph of g(x) is below. Use these to compute f (g(4)) and f (g(−2)). 6 f (x) = 4 |x − 6| + 1, x≤2 1 − x2 , x>2 2 −6 −4 −2 2 4 6 −2 −4 −6 8. For each of the following, write the function given as a composition of functions f (g(x)). r (a) h(x) = (b) j(x) = (3x2 − 4)3 2x − 1 3x + 4 5 9. For each of the following transformations of functions, state a correct order of transformations, then sketch the resulting function. Extra graphs have been included which you may optionally use to show intermediate steps of the sequence of transformations. In both parts, we will transform the function f (x) graphed below. 8 6 4 2 −8 −6 −4 −2 −2 2 4 6 8 −4 −6 −8 (a) g(x) = 2f (−2(x + 3)) 8 8 8 6 6 6 4 4 4 2 2 2 −8 −6 −4 −2 −2 1 (b) h(x) = − f 2 2 4 6 8 −8 −6 −4 −2 −2 2 4 6 8 −8 −6 −4 −2 −2 −4 −4 −4 −6 −6 −6 −8 −8 −8 8 8 8 6 6 6 4 4 4 2 2 2 −8 −6 −4 −2 −2 2 4 6 8 2 4 6 8 1 x −1 2 2 4 6 8 −8 −6 −4 −2 −2 2 4 6 8 −8 −6 −4 −2 −2 −4 −4 −4 −6 −6 −6 −8 −8 −8 6 Exam 1 Review Math 133 10. Write a formula for the absolute value function graphed below. 8 6 4 2 −8 −6 −4 −2 −2 2 4 6 8 −4 −6 −8 11. If b(x) = −2|x + 5| + 6, find the x- and y-intercepts of b(x), and graph the function on the given axes below. 8 6 4 2 −8 −6 −4 −2 −2 2 4 6 8 −4 −6 −8 7 12. Find the inverse of the following functions. (a) f (x) = 9 + 10x (b) g(x) = x x+2 13. Find an equation of the line (a) Parallel to y = 2x + 3 through the point (0, 8) (b) Perpendicular to y = 2x + 3 through the point (−4, 2) (c) Parallel to y = 13 x − 1 through the point (6, 0) (d) Perpendicular to y = 13 x − 1 through the point (1, 1) 8