Download Artificial Intelligence - KBU ComSci by : Somchai

2
3
DEFINITION OF FUNCTION
• THE ELEMENT X OF D IS THE ARGUMENT OF F. THE SET D IS THE
DOMAIN OF THE FUNCTION. THE ELEMENT Y OF E IS THE
VALUE OF F AT X (OR THE IMAGE OF X UNDER F ) AND IS
DENOTED BY F (X), READ “F OF X.”
• THE RANGE OF F IS THE SUBSET R OF E CONSISTING OF ALL
POSSIBLE VALUES F (X) FOR X IN D.
• NOTE THAT THERE MAY BE ELEMENTS IN THE SET E THAT ARE
NOT IN THE RANGE R OF F.
DEFINITION OF FUNCTION
• CONSIDER THE DIAGRAM IN FIGURE 2.
Figure 2
• THE CURVED ARROWS INDICATE THAT THE ELEMENTS F (W),
F (Z), F (X) AND F (A) OF E CORRESPOND TO THE ELEMENTS
W, Z, X, AND A OF D.
DEFINITION OF FUNCTION
• TO EACH ELEMENT IN D THERE IS ASSIGNED EXACTLY ONE
FUNCTION VALUE IN E; HOWEVER, DIFFERENT ELEMENTS OF D, SUCH
AS W AND Z IN FIGURE 2, MAY HAVE THE SAME VALUE IN E.
• THE SYMBOLS
•
F : D  E,
AND
• SIGNIFY THAT F IS A FUNCTION FROM D TO E, AND WE SAY THAT F
MAPS D INTO E. INITIALLY, THE NOTATIONS F AND F (X) MAY BE
CONFUSING. REMEMBER THAT F IS USED TO REPRESENT THE
FUNCTION.
DEFINITION OF FUNCTION
• IT IS NEITHER IN D NOR IN E. HOWEVER F (X), IS AN ELEMENT
OF THE RANGE R—THE ELEMENT THAT THE FUNCTION F
ASSIGNS TO THE ELEMENT X, WHICH IS IN THE DOMAIN D.
• TWO FUNCTIONS F AND G FROM D TO E ARE EQUAL, AND
WE WRITE
•
F = G PROVIDED
F (X) = G(X)
FOR EVERY X IN D.
• FOR EXAMPLE, IF G(X) = (2X2 – 6) + 3 AND F (X) = X2 FOR
EVERY X IN , THEN G = F.
EXAMPLE 1 – FINDING FUNCTION
VALUES
• LET F BE THE FUNCTION WITH DOMAIN
EVERY X IN .
SUCH THAT F (X) = X2 FOR
• (A) FIND F (– 6), F (
), F (A + B), AND F (A) + F (B), WHERE A
AND B ARE REAL NUMBERS.
• (B) WHAT IS THE RANGE OF F ?
• SOLUTION:
• (A) WE FIND VALUES OF F BY SUBSTITUTING FOR X IN THE
EQUATION
F (X) = X2:
•
F (– 6) = (– 6)2 = 36
•
F(
)=(
)2 = 3
EXAMPLE 1 – SOLUTION
cont’d
•
F (A + B) = (A + B)2 = A2 + 2AB + B2
•
F (A) + F (B) = A2 + B2
• (B) BY DEFINITION, THE RANGE OF F CONSISTS OF ALL NUMBERS OF
THE FORM F (X) = X2 FOR X IN .
•
SINCE THE SQUARE OF EVERY REAL NUMBER IS NONNEGATIVE,
THE RANGE IS CONTAINED IN THE SET OF ALL NONNEGATIVE
REAL NUMBERS.
•
MOREOVER, EVERY NONNEGATIVE REAL NUMBER C IS A VALUE
OF F, SINCE F (
)=(
)2 = C. HENCE, THE RANGE OF F IS
THE SET OF ALL NONNEGATIVE REAL NUMBERS.
EXAMPLE 2 – FINDING FUNCTION
VALUES
•
LET G (X) =
•
(A) FIND THE DOMAIN OF G.
•
(B) FIND G(5), G(–2), G(–A), AND –G(A).
•
SOLUTION:
•
(A) THE EXPRESSION
IS A REAL NUMBER IF AND
ONLY IF THE RADICAND 4 + X IS NONNEGATIVE AND THE
DENOMINATOR 1 – X IS NOT EQUAL TO 0.
•
•
THUS, EXISTS IF AND ONLY IF
4 + X  0 AND 1 – X ≠ 0
EXAMPLE 2 – SOLUTION
•
•
•
cont’d
OR, EQUIVALENTLY,
X–4
AND
X ≠ 1.
WE MAY EXPRESS THE DOMAIN IN TERMS OF INTERVALS AS
[– 4, 1)  (1,
).
• (B) TO FIND VALUES OF G, WE SUBSTITUTE FOR X:
EXAMPLE 2 – SOLUTION
cont’d
DEFINITION OF FUNCTION
• GRAPHS ARE OFTEN USED TO DESCRIBE THE VARIATION OF
PHYSICAL QUANTITIES. FOR EXAMPLE, A SCIENTIST MAY USE
THE GRAPH IN FIGURE 5 TO INDICATE THE TEMPERATURE T OF
A CERTAIN SOLUTION
AT VARIOUS TIMES T DURING AN EXPERIMENT.
Figure 5
DEFINITION OF FUNCTION
• THE SKETCH SHOWS THAT THE TEMPERATURE INCREASED
GRADUALLY FROM TIME T = 0 TO TIME T = 5, DID NOT
CHANGE BETWEEN T = 5 AND T = 8, AND THEN DECREASED
RAPIDLY FROM T = 8 TO T = 9.
• SIMILARLY, IF F IS A FUNCTION, WE MAY USE A GRAPH TO
INDICATE THE CHANGE IN F (X) AS X VARIES THROUGH THE
DOMAIN OF F.
• SPECIFICALLY, WE HAVE THE FOLLOWING DEFINITION.
DEFINITION OF FUNCTION
• IN GENERAL, WE MAY USE THE FOLLOWING GRAPHICAL TEST
TO DETERMINE WHETHER A GRAPH IS THE GRAPH OF A
FUNCTION.
• THE X-INTERCEPTS OF THE GRAPH OF A FUNCTION F ARE THE
SOLUTIONS OF THE EQUATION F (X) = 0. THESE NUMBERS ARE
CALLED THE ZEROS OF THE FUNCTION.
• THE Y-INTERCEPT OF THE GRAPH IS F (0), IF IT EXISTS.
EXAMPLE 3 – SKETCHING THE GRAPH OF A
FUNCTION
•
LET F (X) =
•
(A) SKETCH THE GRAPH OF F.
•
(B) FIND THE DOMAIN AND RANGE OF F.
•
SOLUTION:
•
(A) BY DEFINITION, THE GRAPH OF F IS THE GRAPH OF THE EQUATION
•
•
•
Y=
THE FOLLOWING TABLE LISTS COORDINATES OF SEVERAL POINTS
ON THE GRAPH.
EXAMPLE 3 – SOLUTION
• PLOTTING POINTS, WE OBTAIN THE SKETCH SHOWN IN FIGURE 7.
NOTE THAT THE X-INTERCEPT IS 1 AND THERE IS NO Y-INTERCEPT.
Figure 7
EXAMPLE 3 – SOLUTION
• (B) REFERRING TO FIGURE 7, NOTE THAT THE DOMAIN OF F
CONSISTS OF ALL REAL NUMBERS X SUCH THAT X  1 OR,
EQUIVALENTLY, THE INTERVAL [1, ).
•
THE RANGE OF F IS THE SET OF ALL REAL NUMBERS Y
SUCH THAT Y  0 OR, EQUIVALENTLY, [0, ).
DEFINITION OF FUNCTION
• IN GENERAL, WE SHALL CONSIDER FUNCTIONS THAT INCREASE
OR DECREASE ON AN INTERVAL I, AS DESCRIBED IN THE
FOLLOWING CHART, WHERE X1 AND X2 DENOTE NUMBERS IN I.
• INCREASING, DECREASING, AND CONSTANT FUNCTIONS
DEFINITION OF FUNCTION
•
DEFINITION OF FUNCTION
• AN EXAMPLE OF AN INCREASING FUNCTION IS THE IDENTITY
FUNCTION, WHOSE EQUATION IS F (X) = X AND WHOSE
GRAPH IS THE LINE THROUGH THE ORIGIN WITH SLOPE 1.
• AN EXAMPLE OF A DECREASING FUNCTION IS F (X) = – X, AN
EQUATION OF THE LINE THROUGH THE ORIGIN WITH SLOPE –
1. IF F (X) = C FOR EVERY REAL NUMBER X, THEN F IS CALLED A
CONSTANT FUNCTION.
• WE SHALL USE THE PHRASES F IS INCREASING AND F (X) IS
INCREASING INTERCHANGEABLY. WE SHALL DO THE SAME
WITH THE TERMS DECREASING AND CONSTANT.
EXAMPLE 4 – USING A GRAPH TO FIND DOMAIN, RANGE, AND
WHERE A FUNCTION INCREASES OR DECREASES
•
LET F(X) =
•
(A) SKETCH THE GRAPH OF F.
•
(B) FIND THE DOMAIN AND RANGE OF F.
•
(C) FIND THE INTERVALS ON WHICH F IS INCREASING OR IS
DECREASING.
•
SOLUTION:
•
(A) BY DEFINITION, THE GRAPH OF F IS THE GRAPH OF THE EQUATION
•
•
Y=
WE KNOW FROM OUR WORK WITH CIRCLES THAT THE GRAPH OF
X2 + Y2 = 9 IS A CIRCLE OF RADIUS 3 WITH CENTER AT THE ORIGIN.
EXAMPLE 4 – SOLUTION
• SOLVING THE EQUATION X2 + Y2 = 9 FOR Y GIVES US Y = 
• IT FOLLOWS THAT THE GRAPH OF F IS THE UPPER HALF OF THE
CIRCLE, AS ILLUSTRATED IN FIGURE 8.
Figure 8
EXAMPLE 4 – SOLUTION
• (B) REFERRING TO FIGURE 8, WE SEE THAT THE DOMAIN OF F
IS THE CLOSED INTERVAL [–3, 3], AND THE RANGE OF F IS THE
INTERVAL [0, 3].
• (C) THE GRAPH RISES AS X INCREASES FROM –3 TO 0, SO F IS
INCREASING ON THE CLOSED INTERVAL [–3, 0].
•
THUS, AS SHOWN IN THE PRECEDING CHART, IF X1 < X2 IN
[–3, 0], THEN F (X1) < F (X2)
(NOTE THAT POSSIBLY X1 = –3 OR X2 = 0)
EXAMPLE 4 – SOLUTION
cont’d
• THE GRAPH FALLS AS X INCREASES FROM 0 TO 3, SO F IS
DECREASING ON THE CLOSED INTERVAL [0, 3].
• IN THIS CASE, THE CHART INDICATES THAT IF X1 < X2 IN [0, 3],
THEN
F (X1) > F (X2) (NOTE THAT POSSIBLY X1 = 0 OR X2 = 3)
DEFINITION OF FUNCTION
• THE FOLLOWING TYPE OF FUNCTION IS ONE OF THE MOST BASIC IN
ALGEBRA.
THE GRAPH OF F IN THE PRECEDING DEFINITION IS THE GRAPH OF
Y = AX + B, WHICH, BY THE SLOPE-INTERCEPT FORM, IS A LINE WITH
SLOPE A AND Y-INTERCEPT B.
• THUS, THE GRAPH OF A LINEAR FUNCTION IS A LINE.
EXAMPLE 6 – SKETCHING THE GRAPH OF A LINEAR
FUNCTION
• LET F (X) = 2X + 3.
• (A) SKETCH THE GRAPH OF F.
• (B) FIND THE DOMAIN AND RANGE OF F.
• (C) DETERMINE WHERE F IS INCREASING OR IS DECREASING.
• SOLUTION:
• (A) SINCE F (X) HAS THE FORM AX + B, WITH A = 2 AND B = 3, F IS
A LINEAR FUNCTION.
EXAMPLE 6 – SOLUTION
• THE GRAPH OF Y = 2X + 3 IS THE LINE WITH SLOPE 2 AND
Y-INTERCEPT 3, ILLUSTRATED IN FIGURE 10.
Figure 10
EXAMPLE 6 – SOLUTION
• (B) WE SEE FROM THE GRAPH THAT X AND Y MAY BE ANY
REAL NUMBERS, SO BOTH THE DOMAIN AND THE RANGE OF F
ARE
• (C) SINCE THE SLOPE A IS POSITIVE, THE GRAPH OF F RISES AS
X INCREASES; THAT IS, F (X1) < F (X2) WHENEVER X1 < X2.
THUS, F IS INCREASING THROUGHOUT ITS DOMAIN.
SOME STANDARD REAL FUNCTIONS (CONSTANT
FUNCTION)
A function f : R  R is defined by
f  x  = c for all x  R, where c is a fixed real number.
Y
(0, c)
O
f(x) = c
X
Domain = R
Range = {c}
IDENTITY FUNCTION
A function I:R  R is defined by
I  x  = x for all x  R
Y
I(x) = x
Domain = R
450
O
X
Range = R
MODULUS FUNCTION
A function f : R  R is defined by
 x, x  0
f x = x = 
-x, x < 0
Y
Domain = R
Range = Non-negative real numbers
f(x) = x
f(x) = - x
O
X