CalcNotes01

CalcNotes01 - (Chapter 1: Review) 1.01 CHAPTER 1: REVIEW...

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(Chapter 1: Review) 1.01 CHAPTER 1: REVIEW TOPIC 1: FUNCTIONS PART A: AN EXAMPLE OF A FUNCTION Consider a function f whose rule is given by fx () = x 2 . As a short cut, we often say, “the function = x 2 .” Warning : is referred to as “ f of x ” or “ f at x .” It does not mean “ f times x .” x is the input (or argument ) for f , and x 2 is the output or function value . x ± f ± x 2 This function squares its input, and the result is its output. Note : The rule for this function could have been given as: fu = u 2 , for example. Example : f 3 = 3 2 = 9 3 ± f ± 9 We can think of a function as a calculator button . In fact, your calculator should have a “squaring” button labeled x 2 .
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(Chapter 1: Review) 1.02 f is a function , because no “legal” input yields more than one output. There is no function button on a calculator that ever outputs two or more values at the same time. The calculator never outputs, “I don’t know. The answer could be 3 or ± 10 .” Note : A function is a special type of relation . Relations that are not functions permit multiple outputs for a legal input. PART B: POLYNOMIAL FUNCTIONS A polynomial expression in x can be written in the form: a n x n + a n ± 1 x n ± 1 + ... + a 1 x + a 0 , where n is a nonnegative integer called the degree of the polynomial, the a i coefficients are typically real numbers, and the leading coefficient a n ± 0 . A polynomial function has a rule that can be written as: fx () = polynomial in x . Example 4 x 3 ± 5 2 x 2 + 1 is a 3 rd -degree polynomial in x with leading coefficient 4. The rule = 4 x 3 ± 5 2 x 2 + 1 corresponds to a polynomial function f .
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(Chapter 1: Review) 1.03 PART C: RATIONAL FUNCTIONS A rational expression in x can be expressed in the form: polynomial in x nonzero polynomial in x . Examples : 1 x , 5 x 3 ± 1 x 2 + 7 x ± 2 , x 7 + x which equals x 7 + x 1 ± ² ³ ´ µ Observe in the second example that irrational numbers such as 2 are permissible. The last example correctly suggests that all polynomials are rational expressions. A rational function has a rule that can be written as: fx () = rational expression in x . PART D: ALGEBRAIC FUNCTIONS An algebraic expression in x looks like a rational expression, except that radicals and exponents that are noninteger rational numbers such as 5 7 ± ² ³ ´ µ are allowed even when x appears in a radicand or in a base (but not in an exponent). Examples : x , x 3 + 7 x 5/7 x ± x + 5 3 + 4 All rational expressions are algebraic. An algebraic function has a rule that can be written as: = algebraic expression in x .
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(Chapter 1: Review) 1.04 Here’s a Venn diagram for standard symbolic mathematical expressions: PART E: DOMAIN AND RANGE The domain of a function f , abbreviated Dom f () , is the set of all “legal” inputs . The range of f is then the set of all resulting outputs .
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CalcNotes01 - (Chapter 1: Review) 1.01 CHAPTER 1: REVIEW...

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