Review 1. 1 Relations and Functions. Review Problems

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1 Review 1 1 Relations and Functions Objectives Relations; represent a relation by coordinate pairs, mappings and equations; functions; evaluate a function; domain and range; operations of functions. Skills Determine if a relation represents a function; find the domain and range of a function; use the interval notation to represent the domain and range of a function; evaluate a function. Review Problems 1. Determine if the relation represents a function. {(1, 5), (2, 5), (3, 5), (1, 6)} {(x, y), (c, b), (α, δ), (c, 4)} {(Seattle, North), (Phoenix, South), (Warm, Phoenix), (Cold, Seattle)} 2. Find the DOMAIN of each function and write your answer in the interval notation. f(x) = x 2 f(x) = x + 3 x 2 4 f(x) = x f(x) = x 5 + 2x f(x) = x 5 f(x) = 3x 7 x Find the RANGE of each function and write your answer in the interval notation. f(x) = x 2 3 f(x) = x + 6 f(x) = x f(x) = x + 3 f(x) = x 3 4. Evaluate each function at given points. f(x) = x 2 + 6x + 5, find f(0), f( 1), f( x). f(x) = x + 3, find f(0), f(1), f( x). x f(x) = x 3, find f(0), f(4) and f(2x).

2 2 Graphs and Properties of Functions Objectives The definition of the graph; vertical line test; graph and obtain information from the graph; even and odd functions; increasing and decreasing functions; local maximum and local minimum; average rate of change; the library of functions; piecewise function; transformations of functions. Skills Determine if a graph represents a function by applying the vertical line test; obtain information from the graph of a function (function values, domain, range, intercepts, etc.); determine if a point is on the graph of a function; find points on the graph; determine if a function is even, odd or neither by the graph; determine if a function is even, odd or neither algebraically; find increasing and decreasing intervals of a function from its graph; find local maximums and local minimums of a functions from its graph; evaluate a piecewise function; construct a piecewise function for a real problem; find the equation of a function after a sequence of given transformations; find the sequence of transformations that transforms function f to function g; identify the graph of a function after transformations. Review Problems 1. Determine if the graph represents a function. 2. Let f be the function whose graph is given by the following picture, answer all questions and write your answer in the interval notation if applicable.

3 (a) What are f(0), f( 3 ) and f(3)? 2 (b) What is the domain and the range of f? (c) List all intercepts of the graph with the x-axis in coordinate pairs. (d) List all intercepts of the graph with the y-axis in coordinate pairs. (e) For what values of x does f(x) = 4? (f) For what values of x is f(x) > 0? (g) For what values of x is f increasing? (h) For what values of x is f decreasing? (i) For what values of x does f(x) have a local maximum? (j) For what values of x does f(x) have a local minimum? 3. Algebraically determine if the function is even, odd or neither. f(x) = x 2 + x + 1 f(x) = x 3 + x f(x) = x Let f(x) be defined as x + 2 x < 1 f(x) = x 2 1 x 1 x + 2 x > 1 (a) Find f( 2), f( 1), f(0) and f(3).

4 (b) Graph f. (c) Find all local maximums. (d) Determine if f is even, odd or neither from your graph. 5. A cell phone company offers a discount plan as: A baseline fee of $25.00 per month plus $0.25/min for the first 100 minutes, and $0.15 for each minute over 100 minutes. (a) If C is the monthly charge for x minutes, express C as a function of x. (b) Use this function to find the charge of 60 minutes. (c) Use this function to find the charge of 200 minutes. (d) Graph C(x). 6. Let f(x) = x + 5, find the equation of function after each transformation. x 3 (i) Shift right for 2 units. (ii) Then shift up for 1 unit. (iii) Then reflect about the x-axis. 7. Let f(x) = x 2, g(x) = (x 2) Find a sequence of transformations such that g is the transformed function of f after all these transformations. Note: the order of transformations matters! Write the description and the intermediate function of each transformation. For example, The first step, Shift left for 1 unit to get (x + 1) 2. 3 Factoring Polynomials Objectives There won t be problems that explicitly test factoring polynomials, but you re required to remember a number of formulas and be able to factor simple quadratic polynomials. Skills factor difference of two squares, perfect squares and sums and difference of two cubes; factor Ax 2 + Bx + C.

5 Review Problems I. Factor polynomials by removing the common monomial factor. 1. 6x 3 3x 2. 8x 4 + 2x 2 + 4x II. Factor difference of two squares and perfect squares. 1. x x x x + 36 III. Factor cube sums and difference. 1. x x IV. Factor polynomials by grouping. 1. 3x 2 + 6x x x 2 3x + 2x 2 V. Factor quadratic polynomials. 1. 3x 2 + 4x x 2 + 5x x 2 17x + 16

6 4 The Complex Number System Objectives Imaginary unit; complex numbers. Skills Rewrite a complex number in the standard form; adding and subtracting complex numbers; multiplying complex number; calculate the conjugate of a complex number; know properties of conjugates; calculate the reciprocal of a complex number; divide a complex number by another one; calculate the power of the imaginary unit; calculate the principle square root. Review Problems 1. Write the answer of each problem in the standard form: (5 + 4i) + (3 12i) (3 + πi) (12 3πi) (6 + 12i) 7 (3 + 2i)(7 8i) (2 7i)(5 + 6i) ( 3 + i)(i 12) 3 + 4i 5 + 6i 14i 9 2i 2 4i 12i 2. Let z = a + bi be a complex number, which of the following is NOT correct? A. z = z. B. If b = 0, z = z. C. z z = a 2 + b 2. D. z z is a real number. 3. Find powers of the imaginary unit: i 3 i 123 i 99 i 3 (i + 1) (i 2 + 1)(i 9 + i 8 + i 7 + i 6 + i 5 ) 4. Find the principle square roots of 16 and (5 12i)(12i + 5).

7 Selected Answers Answers might not be right. Contact me if you have any questions. Relations and Function 1. no, no, yes. 2. First line: (, ), (, 0] and (, 5]. Second line: (, 2) ( 2, 2) (2, ), (, ) and (, ). 3. First line: [ 3, ), [0, ) and (, ). Second line: (, ) and [3, ). 4. First line: 5, 0 and x 2 6x + 5. Second line: 1, 1 and x + 3 x Third line: 0, 8 and 8x x + h + 2, 4x + 2h and 1. Graphs and Properties of Functions 1. Yes, no. 2. Skipped. 3. Neither, odd, even. 4. Skipped. 5. Skipped. 6. x + 3 x Shift right 2 units, shift up 7 units.

8 Factoring Polynomials I. 1. 3x(2x 2 1) 2. 2x(4x 3 + x + 2) II. (x + 3)(x 3) (5x + 9)(5x 9) (x + 6) 2 III. (x + 1)(x 2 x + 1) (x 4)(x 2 + 4x + 16) IV. (3x 1)(x + 2) (3x + 2)(x 1) V. (x + 1)(3x + 1) (x + 2)(x + 3) (x 1)(x 16) The Complex Number System (1) 8 8i 9 + 4πi i (2) D i 52 23i i 9 38i i 1 1i (3) i i i (4) 4i, 13i

SOLUTIONS FOR PROBLEMS 1-30

SOLUTIONS FOR PROBLEMS 1-30 . Answer: 5 Evaluate x x + 9 for x SOLUTIONS FOR PROBLEMS - 0 When substituting x in x be sure to do the exponent before the multiplication by to get (). + 9 5 + When multiplying ( ) so that ( 7) ( ).