1.definition of a function 2.finding function values 3.using the vertical line test
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1. Definition of a function 2. Finding function values3. Using the vertical line test
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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
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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
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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 )
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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
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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