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MATH 157
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Colin Stewart
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Mathematics

MATH 157

Colin Stewart

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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