Understanding Engineering Mathematics

3.2 Revision

3.2.1 Definition of a function

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Afunctionis a relation which expresses how the value of one quantity, thedependent
variable, depends on the value of another, theindependent variable. Formally, this is
usually expressed
y=f(x)
dependent ↑↑independent
variable variable

wherex is some quantity in a particular set of numbers which, when substituted into
the functionf, produces the quantityy, which may be in a completely different set of
numbers.

Example

Supposex=nis a non-zero integer, then thereciprocal functionis defined by:

y=f(x)=f(n)=

1

n

n= 0

andycan be a rational number of magnitude less than one.
The important point about a function is that it must have a singleuniquevalue,y,for
everyvalue,x, for which the function is defined.
xis also called theargumentof the functionf(x).ThesetXof all values for which the
functionfis defined is called itsdomain. The set of all corresponding values ofy=f(x),
Y, is called therangeoff(x). We sometimes express a function as amappingbetween
the setsX,Ydenotedf:X→Y. The value of a particular function for a particular value
ofx,sayx=a, is called theimageof a underf, denotedf(a).

Examples

(i)y=f(x)=x^2 +1 is defined for all values ofx, and for all such valuesyis greater

than or equal to 1. The image of−1 for this function is

f(− 1 )=(− 1 )^2 + 1 = 2

(ii)y=g(x)=

3

x− 1

is defined for allxexceptx=1, andycan vary over all real

non-zero numbers. The image ofx=0 for example isg( 0 )=

3

0 − 1

=−3.

We have already met the common algebraic functions in Chapter 2:

Linear: f(x)=ax+b (defined for allx)

Quadratic: f(x)=ax^2 +bx+c (allx)

Cubic: f(x)=ax^3 +bx+cx+d (allx)