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Welcome to Fall Semester 2016 Math.1330 – PreCalculus Section 26494 Instructor: Dr. Blerina Xhabli Email: [email protected] Webpage: https://www.math.uh.edu/~blerina Office Location: 609 PGH 1 Important Information to Know: - CourseWare accounts (fingerprints) - Textbook - Daily Poppers - Homework Assignments - Online Quizzes/Practice Tests - 4 Exams and mandatory Final Exam 2 Grades Test 1 Test 2, 3, 4 Final Exam Homework Online Quizzes Daily poppers Note: The percentage grade on the final exam can be used to replace your lowest test score. Course Policy Quiz: You will need to make a 100 before you can access any exams, quiz or homework. 3 Math 1330 – Chapter 1 An Introduction to Functions/Compositions/Inverse Functions Note: This section covers prerequisite material. I will only solve some of the problems here. The rest will be exercises for you… Let A and B be two nonempty sets. A function from A to B is a rule of correspondence that assigns to each element in A exactly one element in B. Here A is called the domain of the function and the set B is called the range of the function. Domain of a Function (the set of all possible inputs) Important Rule: To determine the domain of a function, start with all real numbers and then eliminate anything that results in zero denominators or even roots of negative numbers. Polynomial Functions: the domain is ,, or all real numbers. Rational Functions(both numerator and denominator are polynomials): the domain is all real numbers except the values of x for which the denominator equals 0. Radical Functions with even index: the domain is the set of real numbers for which the radicand is greater than or equal to 0. Radical Functions with odd index: the domain is ,. 4 Example: State the domain of the function. Write your answer using interval notation. a) f ( x) x3 x7 b) g ( x) x2 5x 4 x 2 16 c) h( x) x 4 d) h( x) 3 x 2 9 e) f ( x) x 1 x5 Range of a Function (the set of all possible outputs) Important Rule: To determine the range of a function, determine what outputs are possible. This is not always easy. Sometimes it helps to graph the function. You should know some of these from college algebra. For example: The range of f ( x) x 2 is 0, . The range of g ( x ) x is 0, . The range of h( x) x is 0, . 5 Evaluation of Functions at a given value of x or at an expression. Example: If g ( x) x , find g (1), g (5), g (2 x 1), g (t 1) 2x 4 Piecewise Functions x 1 2 x 4, 2 Example: If f ( x) x 2 x, 1 x 5 , find f (0), f (4), f (5), and f (3). 6 x, x5 Average Rate of Change (Difference Quotient) (you will need this in Calculus!) change in y f ( x h) f ( x) f ( x h) f ( x) change in x ( x h) x h To find a difference quotient, you will compute f ( x h) f ( x ) , assuming that h 0. h You can do this in three steps: 1. Compute f ( x h). 2. Then compute f ( x h) f ( x). 3. Then compute f ( x h) f ( x ) . h 6 Example: Find the difference quotient of: f ( x) x 2 2 x Example: Find the difference quotient at x 2 of: f ( x) 1 x 1 7 Combining Functions We can combine functions in any of five ways. Four of these are the familiar arithmetic operations; addition, subtraction, multiplication and division, and are very intuitive. The fifth type of combining functions is called composition of functions. In all cases, we’ll be interested in combining the functions and in finding the domain of the combined function. Suppose we have two functions, f (x) with domain A and g (x ) with domain B. Let 𝑓(𝑥) = √𝑥 + 1 and 𝑔(𝑥) = 𝑥 − 4 Sum: f g ( x) f ( x) g ( x) Difference: f g ( x) f ( x) g ( x) Product: fg( x) f ( x) g( x) Quotient: f f ( x) ( x) g ( x) g Composition of functions: f g ( x) f ( g ( x)) The domain of f g is the set of all x such that x is in the domain of g and g ( x ) is in the domain of f . 8 Example: Suppose f ( x) 2 x and g ( x) . Find: x 1 2x 4 a) 𝑓(𝑔(2)) b) (𝑓𝑔)(2) c) ( f g )( x ) d) Domain of ( f g )( x ) : e) ( g f )( x ) f) Domain of ( g f )( x ) : Note: ( fg )( x) and f ( g ( x)) have different meanings!!!! Be careful about the notation. 9 Example 3: In the graph below, the function graphed in blue is f (x) and the function graphed in red is g (x). Find each quantity. a. ( f g )(2) b. ( fg )(2) c. f ( g (2)) d. g ( f (2)) e. f (g (3)) f. g ( f (0)) g. f ( f (2)) 10 Inverse Functions A function is one-to-one if no two elements in the domain have the same image. The Horizontal Line Test: A function is one-to-one if any horizontal line intersects the graph of the function in no more than one point. Example 1: Determine if the functions graphed are one-to-one. Example 2: Determine if f ( x) x 2 3 is one-to-one. Example 3: Determine if f ( x) x 3 2 is one-to-one. 11 If a function is one-to-one then there is an associated function called “the inverse”. The inverse function of a one-to-one function is a function f 1 ( x) such that f f 1 f 1 f x. Note: f 1 ( x) 1 f ( x) To determine if two functions are inverses of one another, you need to compose the functions in both orders. Your result should be x in both cases. That is, given two functions f and g, the functions are inverses of one another if and one if f ( g ( x)) g ( f ( x)) x. Example 4: Determine whether f ( x) 5 x 2 and g ( x) x2 are inverses of one another. 5 Note: The inverse function reverses what the function did. Therefore, the domain of f is the range of f 1 and the range of f is the domain of f 1 . Example 5: If f (1) 2, f 1 (1) 0 and f (2) 5, find f (0) and f 1 (5) . 12 The Procedure to find an inverse function: 1. 2. 3. 4. Rewrite the function as y f (x). Interchange x and y. Solve the equation you wrote in step 2 for y. Rewrite the inverse using inverse notation, f 1 ( x). Example 6: You know that f ( x) 4 x 7 is a one-to-one function. Find its inverse. Example 7: Show f ( x) ( x 5) 2 , x 5 is a one-to-one function. Find its inverse. Example 8: f ( x) 1 x is a one-to-one function. Find its inverse. 2 x (Extra) Example 9: Find the inverse of the function f ( x) 5 4 x 1 . 13