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# section07 - Section 7 Functions Functions We now talk about...

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Section 7 – Functions Functions We now talk about functions. MAT 300 Awtrey MATHEMATICS AND STATISTICS 1 / 36

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Section 7 – Functions Functions We now talk about functions. You may think of a function as merely a formula, like MAT 300 Awtrey MATHEMATICS AND STATISTICS 1 / 36
Section 7 – Functions Functions We now talk about functions. You may think of a function as merely a formula, like f ( x ) = x 2 + 3 MAT 300 Awtrey MATHEMATICS AND STATISTICS 1 / 36

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Section 7 – Functions Functions We now talk about functions. You may think of a function as merely a formula, like f ( x ) = x 2 + 3 Thinking of a function in this way is helpful, but inadequate for some things we need to do in analysis. MAT 300 Awtrey MATHEMATICS AND STATISTICS 1 / 36
Section 7 – Functions Functions We now talk about functions. You may think of a function as merely a formula, like f ( x ) = x 2 + 3 Thinking of a function in this way is helpful, but inadequate for some things we need to do in analysis. The general idea of a function will be this: a correspondence between sets where an element in the first set cannot correspond to two different elements in the second set. MAT 300 Awtrey MATHEMATICS AND STATISTICS 1 / 36

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Section 7 – Functions Functions We now talk about functions. You may think of a function as merely a formula, like f ( x ) = x 2 + 3 Thinking of a function in this way is helpful, but inadequate for some things we need to do in analysis. The general idea of a function will be this: a correspondence between sets where an element in the first set cannot correspond to two different elements in the second set. In fact, a function is a special kind of relation. MAT 300 Awtrey MATHEMATICS AND STATISTICS 1 / 36
Section 7 – Functions Functions Definition Let A and B be sets. A function between A and B is a nonempty relation f A B such that if ( a ; b ) 2 f and ( a ; c ) 2 f then b = c . MAT 300 Awtrey MATHEMATICS AND STATISTICS 2 / 36

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Section 7 – Functions Functions Definition Let A and B be sets. A function between A and B is a nonempty relation f A B such that if ( a ; b ) 2 f and ( a ; c ) 2 f then b = c . Definition The domain of f , denoted dom f , is the set of all first elements of members of f . The range of f , denoted rng f , is the set of all second elements of members of f . Symbolically MAT 300 Awtrey MATHEMATICS AND STATISTICS 2 / 36
Section 7 – Functions Functions Definition Let A and B be sets. A function between A and B is a nonempty relation f A B such that if ( a ; b ) 2 f and ( a ; c ) 2 f then b = c . Definition The domain of f , denoted dom f , is the set of all first elements of members of f . The range of f , denoted rng f , is the set of all second elements of members of f . Symbolically dom f = f a 2 A : 9 b 2 B s.t. ( a ; b ) 2 f g rng f = f b 2 B : 9 a 2 A s.t. ( a ; b ) 2 f g The set B is referred to as the codomain of f . If it happens that the domain of f is equal to all of A , then we say f is a function from A into B and we write f : A ! B . It is customary to use the notation f : A ! B only when dom f = A .

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Section 7 – Functions Functions Definition Let A and B be sets. A function between A and B is a nonempty relation f A B
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