Preliminaries About Functions Noel DeJarnette and Austin Rochford August 21, 2012 Navigation Symbols 2, "in" or "an element of", 1 f0; 1; 2; 3; 4g ⊂, "subset of" or "contained in", f0; 1; 2; 3; 4g ⊂ N ⊂ Z ⊂ Q ⊂ R (a; b), interval notation for fx 2 R : a < x < bg,(−∞; 1) is all of R. fx 2 R : x ≥ ag, set notation for [a; 1). N, natural numbers, f1; 2; 3; 4;:::g Z, integers, f:::; −4; −3; −2; −1; 0; 1; 2; 3; 4;:::g n p o Q, rational numbers, x 2 R : x = q where p; q 2 Z; q 6= 0 so f takes pts from a set, A, of the real numbers (it could be all of R Functions, a Doodle The functions we will deal with in Calc 1 (and probably all the functions you have seen and will see until Calc 3) send one real number to another real number. so f takes pts from a set, A, of the real numbers (it could be all of R Functions, a Doodle The functions we will deal with in Calc 1 (and probably all the functions you have seen and will see until Calc 3) send one real number to another real number. Functions, a Doodle The functions we will deal with in Calc 1 (and probably all the functions you have seen and will see until Calc 3) send one real number to another real number. so f takes pts from a set, A, of the real numbers (it could be all of R Functions, a Doodle The functions we will deal with in Calc 1 (and probably all the functions you have seen and will see until Calc 3) send one real number to another real number. so f takes pts from a set, A, of the real numbers (it could be all of R) to another set, B, also in the reals. Functions, a Doodle The functions we will deal with in Calc 1 (and probably all the functions you have seen and will see until Calc 3) send one real number to another real number. so f takes pts from a set, A, of the real numbers (it could be all of R) Functions, a Doodle The functions we will deal with in Calc 1 (and probably all the functions you have seen and will see until Calc 3) send one real number to another real number. so f takes pts from a set, A, of the real numbers (it could be all of R) to another set, B, also in the reals. Functions, a Doodle The functions we will deal with in Calc 1 (and probably all the functions you have seen and will see until Calc 3) send one real number to another real number. so f takes pts from a set, A, of the real numbers (it could be all of R) to another set, B, also in the reals. Functions, a Doodle The functions we will deal with in Calc 1 (and probably all the functions you have seen and will see until Calc 3) send one real number to another real number. More precisely, a number a 2 A Functions, a Doodle The functions we will deal with in Calc 1 (and probably all the functions you have seen and will see until Calc 3) send one real number to another real number. More precisely, a number a 2 A Functions, a Doodle The functions we will deal with in Calc 1 (and probably all the functions you have seen and will see until Calc 3) send one real number to another real number. More precisely, a number a 2 A is sent to a point f (a) = b 2 B Functions, a Doodle The functions we will deal with in Calc 1 (and probably all the functions you have seen and will see until Calc 3) send one real number to another real number. More precisely, a number a 2 A is sent to a point f (a) = b 2 B (meaning f (x) exists) This would be the set A in the doodle. Example The function f (x) = x2 is defined everywhere, so the domain is all of R. Example f (x) = ln x is defined only for strictly positive numbers so the domain is fx 2 R : x > 0g. Domain and Range of Functions Definition The domain of a function f is the set of points where f is defined, . This would be the set A in the doodle. Example The function f (x) = x2 is defined everywhere, so the domain is all of R. Example f (x) = ln x is defined only for strictly positive numbers so the domain is fx 2 R : x > 0g. Domain and Range of Functions Definition The domain of a function f is the set of points where f is defined,(meaning f (x) exists). Example The function f (x) = x2 is defined everywhere, so the domain is all of R. Example f (x) = ln x is defined only for strictly positive numbers so the domain is fx 2 R : x > 0g. Domain and Range of Functions Definition The domain of a function f is the set of points where f is defined,(meaning f (x) exists). This would be the set A in the doodle. Example f (x) = ln x is defined only for strictly positive numbers so the domain is fx 2 R : x > 0g. Domain and Range of Functions Definition The domain of a function f is the set of points where f is defined,(meaning f (x) exists). This would be the set A in the doodle. Example The function f (x) = x2 is defined everywhere, so the domain is all of R. Domain and Range of Functions Definition The domain of a function f is the set of points where f is defined,(meaning f (x) exists). This would be the set A in the doodle. Example The function f (x) = x2 is defined everywhere, so the domain is all of R. Example f (x) = ln x is defined only for strictly positive numbers so the domain is fx 2 R : x > 0g. This would be the set B in the doodle. Example The function f (x) = x2 is always nonnegative, so the range is fx 2 R : x ≥ 0g. Question What does nonnegative mean? How does the domain of ln x differ from the range of x2? Example The function f (x) = ln x gives all values from −∞ to 1 so the range is all of R. Domain and Range of Functions Definition The range of a function f is the set of points that are the values f (x). Example The function f (x) = x2 is always nonnegative, so the range is fx 2 R : x ≥ 0g. Question What does nonnegative mean? How does the domain of ln x differ from the range of x2? Example The function f (x) = ln x gives all values from −∞ to 1 so the range is all of R. Domain and Range of Functions Definition The range of a function f is the set of points that are the values f (x). This would be the set B in the doodle. Question What does nonnegative mean? How does the domain of ln x differ from the range of x2? Example The function f (x) = ln x gives all values from −∞ to 1 so the range is all of R. Domain and Range of Functions Definition The range of a function f is the set of points that are the values f (x). This would be the set B in the doodle. Example The function f (x) = x2 is always nonnegative, so the range is fx 2 R : x ≥ 0g. Example The function f (x) = ln x gives all values from −∞ to 1 so the range is all of R. Domain and Range of Functions Definition The range of a function f is the set of points that are the values f (x). This would be the set B in the doodle. Example The function f (x) = x2 is always nonnegative, so the range is fx 2 R : x ≥ 0g. Question What does nonnegative mean? How does the domain of ln x differ from the range of x2? Example The function f (x) = ln x gives all values from −∞ to 1 so the range is all of R. Domain and Range of Functions Definition The range of a function f is the set of points that are the values f (x). This would be the set B in the doodle. Example The function f (x) = x2 is always nonnegative, so the range is fx 2 R : x ≥ 0g. Question What does nonnegative mean? How does the domain of ln x differ from the range of x2? Domain and Range of Functions Definition The range of a function f is the set of points that are the values f (x). This would be the set B in the doodle. Example The function f (x) = x2 is always nonnegative, so the range is fx 2 R : x ≥ 0g. Question What does nonnegative mean? How does the domain of ln x differ from the range of x2? Example The function f (x) = ln x gives all values from −∞ to 1 so the range is all of R. The Preimage is a perfect example. It will be used to find inverses of functions and is important in finding the domains of compositions. Domain and Range of Functions Sometimes we want information about subsets of the range or the domain.