MTH 06 LECTURE NOTES (Ojakian) Topic 2: Functions


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1 MTH 06 LECTURE NOTES (Ojakian) Topic 2: Functions OUTLINE (References: Iyer Textbook  pages 4,6,17,18,40,79,80,81,82,103,104,109,110) 1. Definition of Function and Function Notation 2. Domain and Range of Function 3. Special Functions: (a) Polynomial (b) Rational (c) Radicals and rational exponents (d) Exponential and Logarithmic 4. Inverse Functions 1. Functions Definition 1. Let D be some specified set of numbers. A function with domain D is a rule that assigns a unique value to each element of D. PROBLEM 1. For each item state whether or not it is a function. (a) Let D be the set of integers. We define a function which assigns to each integer the result of multiplying it by two and then adding one. (b) We define a function whose domain is the integers and assigns to each integer n, the value 2n + 1. (c) Let G be the function whose domain is the integers and assigns to each integer x, the value 2x + 1. (d) Let f be the function whose domain is the real numbers and assigns to each number u the value 3/u. (e) Let F be the function which assigns to each integer y, the number u such that u 2 is y. (f) Let g be the function whose domain is the positive integers and assigns the even integers to 5 and and odd integers to 2. *PROBLEM* 2. For each item state whether or not it is a function. (a) Let S be the set of positive real numbers. We define a function which assigns to each number in the domain, half its value. (b) Let f be the function which assigns the rational numbers to 1 and the integers to 1. (c) Let g be the function with domain being all real numbers, which assigns to each number its square root. 1
2 (d) Let F be the function whose domain is the negative real numbers and assigns to each number x, the number x + 3. (e) Let H be the function whose domain is all the nonzero rational numbers, and assigns to any rational number its reciprocal. Some terminology: Input: Element of Domain to which we apply the function. Value of a function on some input: The result of applying the function to an element of the domain. Output: Same as Value. Write down in a nonnumeric function. 2. Function Notation Definition 2. We can define a function using function notation by: (a) Specifying its domain, and (b) Giving the function a name (like F ), and (c) Describing the rule as follows: F (x) = [PUT RULE DESCRIPTION]. Given an element u in the domain, we write F (u) to refer to the value of F on u. PROBLEM 3. Describe the functions from Problem 1 using function notation. *PROBLEM* 4. Describe the functions from Problem 2 using function notation. 3. Function Evaluation PROBLEM 5. (a) Let f(x) = 3 + x and let the domain be all the nonnegative real numbers. Find the value f(9). Evaluate f(100). (b) Let g(x) = 3 and let the domain be all the integers larger than x Find g(16). Find g(4) (watch it!). *PROBLEM* 6. (a) Let h(y) = 10y 2 + y + 3 and let the domain be all the real numbers y 1. Find the value h( 2). Evaluate h(1). (b) Let H(x) = 10 2 x + 1 and let the domain be all the real numbers less than 9. x 10 Find H(0). Calculate H(4). Give your nonnumeric function a name and use function notation to evaluate it for some inputs. Hand this in. 2
3 4. Domain and Range Properly speaking, when a function is defined, a domain should be specified. However... Definition 3. If the rule for a function is given, but no domain is given, then it is assumed that the domain is the largest set of real numbers for which the rule makes sense. PROBLEM 7. (a) Define a function f(x) = x. What is its domain? x (b) What is the domain of the function g(x) = x 7? PROBLEM 8. (a) Define a function g(u) = 2 u. What is its domain? (b) What is the domain of the function g(x) = 3 x (x + 7)(x 5)? Definition 4. The range of a function is the set of all possible values of the function. PROBLEM 9. Find the range of each function. (a) Consider the function f(y) = 2y where the domain is the positive integers. (b) Consider the function f(u) = 2u. (c) Consider the function g(x) = 3 x and its domain is the set { 2, 1, 0, 1, 2}. *PROBLEM* 10. Find the range of each function. (a) Consider the function f(y) = 2y + 1 where the domain is the nonnegative integers. (b) Consider the function f(u) = u. (c) Consider the function h(x) = 1 and its domain is the set { 2, 1, 0, 1, 2}. 3x (d) Give an example of a function that has 3 elements in its range, but infinitely many elements in its domain. 5. Constants Definition 5. In a discussion, if we refer to a symbol as a constant, we mean that the symbol represents one fixed real number for the entire discussion. PROBLEM 11. Consider a constant k. For which values of x is the expression undefined? 1 x k PROBLEM 12. Suppose c is a positive integer constant larger than 1. Suppose we define the function f(x) = c x. For which choices of c is f defined everywhere? *PROBLEM* 13. Consider a constant k. For which values of x does the expression x + k x 2 evaluate to zero? 6. Polynomials Expressions and Functions 3
4 Definition 6. A monomial in the variable x is an expression of the form c x n where c and n are constants such that n is a nonnegative integer. A monomial in the variables x, y is an expression of the form c x n y m where c, n, and m are constants such that n and m are nonnegative integers. *PROBLEM* 14. (a) Give a definition of a monomial in variables x, y, z. (b) Give a definition of a monomial in general. Hint: Use variables x 1, x 2,..., x v and constants n 1, n 2,..., n v. Definition 7. A polynomial is a sum of monomials. Definition 8. A polynomial function is a function whose rule is defined using a polynomial expression. PROBLEM 15. Suppose p(x, y) = x 177 y 22 x 27 y p( 1, 1). Evaluate p(1, 1) and PROBLEM 16. Suppose g(x) = 2x 2 and h(x) = x/ Evaluate g(h(10)) and h(g(10)). *PROBLEM* 17. (a) Consider the polynomial expression 3x 3 x 2 2x 1. Evaluate the expression when x = 10, when x = 10, and when x = 1/2. (b) Suppose f(x) = 20x 34 x 3. What is the domain of f? Evaluate f(0), f(1), and f( 1). (c) Suppose g(x) = x 2. What is the domain and range of g? (d) Suppose f(x) = x 2 1 and g(x) = x + 2. Evaluate f(g(1)). Evaluate g(f(1)). (e) Find a polynomial function p(x, y) so that for any real numbers a and b, p(a, b) and p(b, a) have the same value. (f) Suppose p is any polynomial function. What, if anything, can you say about its domain and about its range? 7. Rational Expressions and Functions Definition 9. A rational expression is an expression of the form P Q are both polynomials. where P and Q Definition 10. A rational function is a function whose rule is defined using a rational expression. PROBLEM 18. Consider the rational expression y Evaluate it when y = 0, y + 7 when y = 7, and when y = 7. *PROBLEM* 19. (a) Consider the rational function R(u) = u 5 u 2. Evaluate R(0), R( 5), and R( 2). 4 What is the domain of R? 4
5 (b) Define a rational function whose domain is all real numbers. (c) Define a rational function whose value is never zero. 8. Rational Exponents Property 1. Suppose x is a real number. Suppose p and q are integers such that q 0. x 0 = 1 x q = 1 x q x p/q = q x p = ( q x) p PROBLEM 20. (a) Let f(x) = 10x 2/3. Evaluate f(27) and f( 27). What is the domain of f? (b) Consider the function f(x) = x 1/2 and g(x) = x 2. Evaluate f(g(5)) and g(f(4)). *PROBLEM* 21. (a) Let G(x) = 8x 1/2. Evaluate G(9) and G( 9). What is the domain of G? (b) Write the expression 4 x 3 using exponents. (c) Let f(x) = x 5/3 and let g(x) = x 5/2 Which is larger, f(2016) or g(2016)? (d) Consider the function f(x) = x 1/100 and g(x) = x 100. Evaluate f(g(7)) 9. Inverses Definition 11. Consider two functions f and g. They are inverses if the following two conditions hold: For all real numbers x in the domain of g: f(g(x)) = x. For all real numbers x in the domain of f: g(f(x)) = x. We say that f is the inverse of g and that g is the inverse of f. Definition 12. If a function is named F, then we use F 1 function. to refer to its inverse PROBLEM 22. (a) Let f(x) = x + 5 and g(x) = x 5. Verify that f and g are inverses. (b) Let h(x) = 2x. What is the inverse of h? *PROBLEM* 23. (a) Let f(x) = 5x and g(x) = x/5. Verify that f and g are inverses. (b) Let h(x) = 2x + 1. What is the inverse of h? Aside: Not all functions have inverses. But we won t worry about that! PROBLEM 24. Consider any of the above pairs of inverses. Compute some values of each function in order to make an inputoutput table. Do you notice a relationship? 5
6 *PROBLEM* 25. Do Problem 24 again, but for a different pair of inverse functions from above. Do you notice a relationship? *PROBLEM* 26. Suppose g and h are inverse functions. Suppose g(2) = 3 and g(0) = 9. What can you conclude about h? Why? In order to calculate h(17), what would you need to know about g? 10. Exponential Functions and Logarithmic Functions *PROBLEM* 27. Let b be a constant. Define a function f(x) = b x. Consider what happens for various fixed values of the constant b: (a) Consider the case in which b = 1. Calculate f(999). Describe f(x). Do the same for the case of b = 0. (b) Compare the case of b = 2 versus the case of b = 2. In one case, the domain will be all real numbers. In the other case, the domain will be missing lots of numbers. Figure out which case is which. Hint: Look back at Problem 12. Definition 13. Let b be a real number constant such that b 1 and b > 0. We define the function exp b, which is called the exponential function with base b. It is defined by: exp b (x) = b x Definition 14. We define log b to be the inverse of the function exp b. PROBLEM 28. Evaluate the following: log 2 (8), log 2 (4), log 2 (1/4). *PROBLEM* 29. Evaluate the following: log 3 (9), log 3 (1/9), log 3 (3), log 3 (1). *PROBLEM* 30. Evaluate the following (there is a short way and a loooong way). (a) log 7 (exp 7 (3)) (b) exp 2 (log 2 (64)) (c) log 4 (4 19 ) (d) 9 log 9 (81) 11. Final Problems *PROBLEM* 31. (a) Suppose k is a constant real number. Define the function F (x) = x + k. What is the inverse function of F? (b) Suppose a and b are nonzero constants. Define the function G(x) = ax + b. What is the inverse function of G? *PROBLEM* 32. Which of the following propositions are theorems? answers Justify your (a) f(3) = 2 f 1 (2) = 3 (b) f(3) = 2 f 1 (2) = 3 (c) f(3) = 2 f 1 (2) = 3 (d) If a, b, and c are real constants then the expression (x a)(x b)(x c) evaluates to zero for three different real numbers choices for x. 1 (e) Suppose a and b are real constants and f(x) = (x a)(x b). a b the domain of f includes all real numbers except for two of them. 6
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