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School
Department
Mathematics
Course
MATH 157
Professor
Colin Stewart
Semester
Fall

Description
Note on Functions and Limits If there are two variables (usually denoted by x and y or s and t) related to each other, then a function is the object to tell us how these two variables related. Deﬁnition 1. A function f is a rule that assigns (maps) each element in an set A to one and only one element in an set B. Remarks: 1. Every element in A must has a unique image f(x), i.e. x can’t be mapped to more than 1 image (Vertical Lines Test). 2. Element in B may not be an image of x in A. Note that f is a function and f(x) is the value of the images of x under f. 1 E.g. 1. Let f(x) = x + 1 (rule) f (▯1;1) ▯! (▯1;1) x ▯! x + 1 Therefore 2 1 maps to f(1) = 1 + 1 = 2 0 maps to f(0) = 0 + 1 = 1 Notation. Let y = f(x) be a function. Then x is called the independent variable and y is called the dependent variable. We can use other variables to represent the same function. For example, in E.g. 1 the function can also be represented by 2 g(s) = s + 1: 2 2 Both f(x) = x + 1 and g(s) = s + 1 deﬁne the same function. Deﬁnition 2. Domain = the set of all x in A where f(x) is deﬁned. = fx 2 A : f(x) is deﬁned g where ”x 2 A” means ”x belongs to the set A”. Range = the set of all images f(x) in B of x in A. = fy 2 B : y = f(x) for some x in A g: From now on, A and B are (▯1;1). p E.g. 2. Let f(x) =x ▯ 1. Then ▯ p Domain = x 2 A : f(x) = x ▯ 1 is deﬁned = fx 2 A : x ▯ 1 ▯ 0g = fx 2 A : x ▯ 1g = [1;1): The range R is more difﬁcult to ﬁnd. We can ﬁnd that f ▯ p R f y : y = f(x) = x ▯ 1 for some x in [1;1) = [0;1): 2 3 Deﬁnition 3. The graph of a functions the set of all points (x;f(x)) in X ▯ Y plane for all x in the domain. Algebra of Functions Let f and g be functions with domains A and B respectively. Then we can construct new functions by the operators +;▯;▯;▯ from f and g. Deﬁnition 4. The sum function of f and g denoted by f + g is deﬁned by (f + g)(x) := f(x) + g(x); for all x 2 A \ B: 4 Here ”:=” means ”deﬁnes” and ”x 2 A \ B” means ”x belongs to A and B”. Also the difference, product and quotient function of f and g are denoted and deﬁned by (f ▯ g)(x) := f(x) ▯ g(x); for all x 2 A \ B: (f ▯ g)(x) := f(x)g(x); for all x 2 A \ B: and ▯ ▯ f (x) :=f(x); for all x 2 A \ B and g(x) 6= 0: g g(x) Remark. The order of f and h in these new functions may matters. For example, in general, (f ▯ g) 6= (g ▯ f) and ▯ f ▯ g 6= : g f But it is always true that (f + g) = (g + f) and (f ▯ g) = (g ▯ f): Why? 5 Composition of Functions Let f and g be functions with domains D andfD respegtively. We can construct a new function by applying the function f ﬁrst and then g second. Deﬁnition 5. The composition of function is deﬁned by (g ▯ f)(x) := g (f(x)): i.e. apply f ﬁrst and then g Therefore, under g ▯ f, x maps to g(f(x)). 6 Similarly, (f ▯ g)(x) := f (g(x)): i.e. apply g ﬁrst and then f Therefore, under f ▯ g, x maps to f(g(x)). Remark. We must assume that f(x) belongs to D in g gf and g(x) belongs to D in f ▯f. Otherwise the composition of f and g are not deﬁned. In general, (g ▯ f) 6= (f ▯ g): Example 1. Let x f(x) = 2 x + 1 and 1 g(x) = : x 7 Note that Df= (▯1;1) and D = (g1;1)nf0g. Then ▯ ▯ x (g ▯ f)(x) = g (f(x)) = g x + 1 2 ▯ 1 ▯ x + 1 = x = x ; for x 6= 0 x +1 and ▯ ▯ 1 (f ▯ g)(x) = f (g(x)) = f x 1 1 1 = ▯ ▯x = x = x 1 2+ 1 12+ 1 1+22 x x x 1 x2 x = ▯ = ; for all x: x 1 + x2 1 + x2 Example 2. Let p f(x) = 2 x + 3 and 2 g(x) = x + 1: Note that Df= [0;1) and D =g(▯1;1). Then ▯ p ▯ (g ▯ f)(x) = g (f(x)) = g 2 x + 3 ▯ p 2 p = 2 x + 3 + 1 = 4x + 12 x + 10; for all x ▯ 0 and ▯ 2 ▯ (f ▯ g)(x) = f (g(x)) = f x + 1 p = 2 x + 1 + 3; for all x: Again g ▯ f 6= f ▯ g. Example 3. Let f(x) = x5 and 1 g(x) = x : Note that Df= D =g(▯1;1). Then ▯ 5▯ (g ▯ f)(x) = g (f(x)) = g x ▯ ▯1 = x5 5 = x; for all x and ▯ 1▯ (f ▯ g)(x) = f (g(x)) = f x 5 ▯ 1▯5 = x5 = x; for all x: 8 In this case, (g ▯ f)(x) = (f ▯ g)(x) = x. Remark. In the last example, under f ▯ g, x maps to itself x. under g ▯ f, x maps to itself x. So f a
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