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Lecture 1

# 55-140 Lecture 1: LawsOfFunctionsExam

Department
Biological Sciences
Course Code
BIOL 1111
Professor
Animesh Sarker
Lecture
1

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Diﬀerential Calculus
MTH 62-140
Some Topics in Functions
1. Function: A function fis a rule that assigns to each element xin a set Dexactly one
element, called f(x), in a set E. The set Dis called the domain of the function. The
range of fis the set of all possible values of f(x) as xvaries throughout the domain.
2. Four ways to represent a function: verbal, visual, numerical and algebraic.
3. Properties of a function:
(a) Domain and range
the set of values for which the function is deﬁned
the set of all possible values of f(x) as xvaries throughout the domain
(b) Intercepts(xand yintercepts):
To ﬁnd xintercept(s), set y= 0 and solve for x.
To ﬁnd yintercept(s), set x= 0 and solve for y.
(c) One-to-one:
A function fis one-to-one if f(x1) = f(x2) implies x1=x2.
A function is one-to-one if and only if no horizontal line intersects its graph
more than once.
(d) Symmetry:
A function fis called an even function if f(x) = f(x) for every x
A function fis called an odd function if f(x) = f(x) for every x
A function fis periodic if f(x) = f(x+p) for some p
(e) Intervals of increase and decrease
A function fis called increasing on an interval Iif f(x1)< f(x2) whenever
x1< x2in I
A function fis called decreasing on an interval Iif f(x1)> f(x2) whenever
x1< x2in I
A function fis periodic if f(x) = f(x+p) for some p
(f) Local minimum and maximum
A function fhas a local maximum at cif f(c)f(x) when xis near c.[This
means that f(c)f(x) for all xin some open interval containing c.]
Afhas a local minimum at cif f(c)f(x) when xis near c.
(g) Absolute minimum and maximum
A function fhas an absolute maximum at cif f(c)f(x) for all xin D,
where Dis the domain of f. The number f(c) is called the maximum value
of fon D.