Math 1314 Lesson 1: Prerequisites

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1 Math 131 Lesson 1: Prerequisites Prerequisites are topics you should have mastered before you enter this class. Because of the emphasis on technology in this course, there are few skills which you will have to do by hand. This lesson is intended as a quick review of these topics. 1. Eponents Recall a couple of Eponent Rules we ll use in this course: n 1 m n n m = = n Eample 1: Simplify and write the answer without using negative eponents: a. 5 b. ( ) 5 Eample : Write as a radical: 3 Eample 3: Write using a rational eponent: 3 5. Analyzing a Polynomial and their Graphs To begin with, let s review the definition of a polynomial function. A polynomial is the sum and/or difference of terms that contain variables and/or real constants, with variables raised to whole number (0, 1,, 3, ) powers. Eample : Which of the following are polynomial functions? 3 a. f ( ) = h. f ( ) = 3e b. f ( ) = 3 i. f ( ) = 5 1 c. f ( ) = 10 d. e. f. g. 5 3 f ( ) = f ( ) = e f ( ) = + f ( ) = ln( 3) Lesson 1 Prerequisites 1

2 The solution(s) (root(s), zero(s), -intercept(s)) of a polynomial function f ( ) is/are found by finding the values of the variable when f ( ) = 0. Eample 5: Find the roots of the function: a. f ( ) = 3. b. ( ) = f To find the y-intercept of a polynomial function, calculate f(0). In fact, to calculate the y- intercept of ANY function, find f(0). Eample 6: Find the y-intercept of f ( ) = 3. Other times, we may need to evaluate a function at a given value. Eample 7: Let 3 ( ) 10 f = +, find f(-). Eample 8: Suppose the total cost in dollars to produce items is given by the function 3 C( ) = Find the total cost of producing 50 items. Lesson 1 Prerequisites

3 The graph of a polynomial function looks similar to one of the four graphs below. Notice that the graphs are nice smooth curves, no sharp corners, no holes and no asymptotes. The domain (set of valid inputs for the function) of any polynomial function is (, ), which we can see clearly from the graphs (the set of -values on the graph). The range (set of outputs after evaluating with valid inputs) is best found by observing the graph (the set of y-values on the graph). The end behavior of a graph is the behavior to the far left and far right of the graph. If we are given only the polynomial function and not its graph, we can determine the end behavior by simply looking at its leading term (term with the highest power on the variable ). End Behavior of a Polynomial Function An even-degree polynomial s end behavior will be if its leading coefficient is negative. An odd-degree polynomial s end behavior will be if its leading coefficient is negative. if its leading coefficient is positive and if its leading coefficient is positive or Eample 9: Determine the end behavior of Eample 10: Determine the end behavior of 5 f ( ) 3 = +. f ( ) ( 1) =. Description of the Behavior at Each -intercept 1. Even Multiplicity: The graph touches the -ais, but does not cross it (looks like a parabola there).. Odd Multiplicity of 1: The graph crosses the -ais (looks like a line there). 3. Odd Multiplicity greater than or equal to 3: The graph crosses the -ais and it looks like a cubic there. Eample 11: Given graph. Leading Term: End Behavior: f ( ) ( 1) ( 1) 3 = + the zeros are 0, 1, and -1, respectively. Sketch its Lesson 1 Prerequisites 3

4 Other times we re given the graph of a polynomial function and wish to find certain function values or when the function is positive or negative. Eample 1: Given the following graph of a polynomial function, a. For which -value(s) is the function equal to 0. b. Find f(-1). c. Find f(). Eample 13: Given the following graph of a polynomial function, a. Give the inteval(s) over which the function is negative. b. Over how many intervals is the funciton positive? c. Give the interval(s) over which the function is positive. d. Find the domain and range. Lesson 1 Prerequisites

5 3. Simplifying an Algebraic Epression 3 + Eample 1: Simplify ( ) ( 3 ) Eample 15: Multiply ( 1)( + 3). Rational Functions If we form the ratio of two polynomials we obtain a rational function. Graphs of rational function may look like: These types of graphs may have zeros, a y-intercept, vertical and/or horizontal asymptotes, and/or holes. The domain of such a function will be the set of all real numbers ecept those - values at any holes and at any vertical aysmptotes. A vertical line is a vertical asymptote of a rational funciton if its graph approaches that line at the far top and far bottom of the graph. The graph can never cross these lines. A horizontal line is a horizontal asymptote of a rational funciton if its graph approaches that line at the far left and far right of the graph. The graph may cross this line. Eample 16: For the graph above, find: a. Any roots. b. Any vertical asymptote. c. Any horizontal asymptote. Lesson 1 Prerequisites 5

6 We can algebraically find a rational function s domain, range, roots, y-intercept, asymptotes, and holes. Recall that we stated before, a rational funciton can never cross its vertical asymptote(s). We shall now see why. Vertical Asymptotes To find a rational function s roots, vertical asymptotes and holes you must first factor the numerator and denominator as much as possible and simplify. Then, LOOK at the denominator. If a factor cancels with a factor in the numerator, then there is a hole where that factor equals zero. If a factor does not cancel, then there is a vertical asymptote where that factor equals zero. The roots are found by setting the numerator equal to zero and solving for. The y-intercept is found by calculating f(0), if possible. The domain of a rational function ecludes those values of where holes and vertical asymptotes occur. (Or set the original function s denominator equal to zero and solve for.) Eample 17: Let g( ) = Find: a. Any vertical asymptote. b. Any roots. c. Any y-intercept. Eample 18: Find the domain of f ( ) =. 3 Lesson 1 Prerequisites 6

7 Horizontal Asymptotes p( ) Let f ( ) =, q( ) Shorthand: degree of f = deg(f), numerator = N, denominator = D 1. If deg(n) > deg(d) then there is no horizontal asymptote.. If deg(n) < deg(d) then there is a horizontal asymptote and it is y = 0 (-ais). 3. If deg(n) = deg(d) then there is a horizontal asymptote and it is y = b a, where a is the leading coefficient of the numerator. b is the leading coefficient of the denominator. Eample 19: Find any horizontal asymptote given: a. g( ) = b. g( ) = Simplifying More Algebraic Epressions Eample 0: Simplify a. ( + + ) b. ( 9 6 ) 3 Lesson 1 Prerequisites 7

8 6. Piecewise Defined Functions A function that is defined by two (or more) equations over a specified domain is called a piecewise function. Eample 1: Let Find: a. f (5). 3 1, < 1 f ( ) = + 5, 1 < 3. + e, 3 b. f ( ). The graphs of piecewise functions look similar to the following graph: Eample : Use the graph above to find each of the following. a. f(0) = b. f(3) = c. f(6) = d. f() = e. For which -value(s) is f() = 3. Lesson 1 Prerequisites 8

9 The last type of function whose domain we need to review is the square root function. Recall that over real numbers we cannot take an even root of a negative number. Hence, to find its domain, eclude real numbers that result in an even root of a negative number. Eample 3: Find the domain and write the answer in interval notation: f ( ) = 3 + Now take Practice Test 1 (0 attempts), then Test 1 ( attempts, from anywhere online, no CASA reservation needed). Lesson 1 Prerequisites 9