1. Definition of a function 2. Finding function values3. Using the vertical line test

Function: A function f is a correspondence from a set D to a set E that assigns to each element x of D exactly one value ( element ) y of E

Graphical Illustration

E

x *

z *

w *

5 *

* f(w)

* f(x)

* f(z)

* f(5)

* 3* 4* - 9

D

f

f is a function

More illustrations….

x *

z *

w *

5 *

* f(w)

* f(x)

* f(z)

* f(5)

* 3* 4* - 9

D

E

f is not a function Why?

x in D has two values

x *

z *

w *

5 *

* f(w)

* f(x)

* f(z)

* f(5)

* 3* 4* - 9

D

E

f is not a function Why?

x in D has no values

How to find function values?

Example 1: Let f be the function with domain R such that f( x) = x2 for every x in R.

( i ) Find f ( -6 ), f ( ), f( a + b ), and f(a) + f(b) where a and b are real numbers.3

Solution: 3666 2 f

3332f

222 2 babababaf

22 babfaf Note: f ( a + b ) f( a ) + f ( b )

Vertical Line Testof functions

Vertical Line test: The graph of a set of points in a coordinate plane is the graph of a function if every vertical line intersects the graph in at most one point

Example: check if the following graphs represent a function or not

Function

Function

FunctionNot Function

Increasing, Decreasing and Constant Functions over an interval I

f(x1) = f(x2)

whenever

x1 x2

f is constant

over interval I

f(x1) > f(x2)

whenever

x1 < x2

f is decreasing

over interval I

f(x1) < f(x2)

whenever

x1 < x2

f is increasing

over interval I

Graphical InterpretationDefinitionTerminology

x1 x2

f(x1)f(x2)

x

y

x1 x2

f(x1) f(x2)

x

y

x1 x2

f(x1) f(x2)

x

y

Interval I

Interval I

Interval I