M1410104 - (Section 1.4: Functions) 1.18 SECTION 1.4:...

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(Section 1.4: Functions) 1.18 SECTION 1.4: FUNCTIONS (See p.40 for definitions of relations and functions and the Technical Note in Notes 1.24 .) Warning: The word “function” has different meanings in mathematics and in common English. Unless otherwise specified, f , g , and h are assumed to be functions. PART A: EXAMPLE Consider a function f whose rule is given by f x ( ) = x 2 . As a short cut, we often say, “the function f x ( ) = x 2 .” Warning: f x ( ) 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: f u ( ) = u 2 , for example. Example: f 3 ( ) = 3 ( ) 2 = 9 3 f 9
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(Section 1.4: Functions) 1.19 Example: f 3 ( ) = 3 ( ) 2 = 9 Warning: It’s often a good idea to use grouping symbols when you make a substitution (or “plug in”). Note that f 3 ( ) is not equal to 3 2 , which equals 9 . Remember that, in the absence of grouping symbols, exponentiation precedes multiplication (by 1 here) in the order of operations. We can think of a function as a calculator button . In fact, your calculator should have a “squaring” button labeled x 2 . 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.
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(Section 1.4: Functions) 1.20 PART B: SOME WAYS TO REPRESENT A FUNCTION A function may be represented by … 1) an algebraic statement such as f x ( ) = x 2 . 2) a description in plain English such as: “This function squares its input, and the result is its output.” 3) an input-output machine such as: x f x 2 4) a table of input x and output f x ( ) values. Although it is often impossible to write a complete table for a function f , a partial table can be useful to look at, especially for graphing. If f x ( ) = x 2 , we can write: x f x ( ) 2 4 1 1 0 0 1 1 2 4
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(Section 1.4: Functions) 1.21 5) a set of input x , output f x ( ) ( ) ordered pairs . For example, the table in 4) yields ordered pairs like 2,4 ( ) . 6) a graph of points that correspond to the ordered pairs in 5); see Section 1.5 . Here is a graph of f x ( ) = x 2 : 7) an arrow diagram such as the one on p.40. 8) an algorithm . Algorithms are discussed in Discrete Math: Math 245 at Mesa . 9) a series . In Calculus: In Calculus II: Math 151 at Mesa , you will use series to represent functions. We will discuss series in Chapter 9 . For example, the function f x ( ) = 1 1 x can be represented by the infinite series 1 + x + x 2 + x 3 + ... , provided that 1 < x < 1 . In Calculus, you will consider series for sin x and cos x , among others.
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M1410104 - (Section 1.4: Functions) 1.18 SECTION 1.4:...

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