Section 3.1-3.2 Functions and Their Graphs
MAC 1105 Sandefur
Objectives: 1) Determine whether a relation represents a function 2) Find the value of a function 3) Find the domain of a function defined by an equation 4) Form the sum, difference, product and quotient of two functions
Defintions: Relation--A set of ordered pairs where the first component (x) corresponds to a second component (y)
Example 1: S ={(1,2), (3,4), (5,6)}
The x value is referred to as the input of the relation and the y value serves as the output. Domain--the set of all first components (x, inputs) Range--the set of all second components (y, outputs) J-xI+I8 Example 2:
a)Find the domain and range of S ={(1,2), (3,4), (5,6)} 0:E1.353 R:EzN.63 b) Create a map (the technique is called mapping) of the relation S. R
Function--a relation where each element of the domain x corresponds to exactly one element of the range y.
Example 3: Consider the following relations. Does each represent a function?
(ndv:7 SDMain Gas Station
( ndo7 Price per Gallon
b) Relation S={(22,3),(4,21),(0,2),(4,25)}
a)
Express the domain and range in a map
Valero
$3.19
7ot a -fUnotion ir there is a duplicate X dupiicate y is Sinel
Shell
$3.29 $3.35
Texaco
Citgo
c) The data expressed in the table:
Figure7 Unleaded price per gallon yes!
16
d) Domain: The set of all US Presidents. Range: Years in which there was a presidential election. nol a funution -range of yevmS
Vertical Line Test--If a vertical line can pass through the graph more than once, the graph is not a function
Example 4: Determine if the graph or equation represents a function. YES or NO?
MAL FLOAT AUTO REAL RADIAN M
b)
hr18 6nce
nits iwice
Equation: ^soh
Equation: No
Equation: yes!
d)
2 +
e) 3 = +1 3J3zJx+i yes1 I+x5=h
f)
= 2 + 3
Jyji6-x2 No y=Ji6-x
yes!
((x-3)(3x+5)
Why is it important to recognize functions??
(x)F<R
8+_(x-)Z- Zx3-8
Function Notation:
E+ ZZz=( )y'sZ e=( )I+ ZZ=( ) :suo!uny8ulMoIoyaytJaprsuoy:sadwex3 Evaluate each of the following: a) f(-2)=-2(z)+1 b) g(3)=J3(3).s c) h(-1)=(-13-z(()+3 =4+1=5 =j=2 =6
d) 3. ( )3[-2x+] =-6x+3
e) g(5x)=J3(sx)-5 =JiSx-S
f) h(x+1) =(x+1)-z(x+1)+ =x3+2
XJF=
Y=fcxJ e:Gll poSsibIe ouHpu1 valUeS
Domain:all possible input ualues Example 6: Find the domain of each of the following functions. Express the domain in set notation. Hint: Avoid division by zero and negative numbers under a square root. a) ()= 2+5 b) (= x34 O O:(-00,00) se {x|xe1R} Ex|x-2,23 IN x&+z c) h( )=V224 d) 0<s-x E+|eEz3 x>S 2-4t70 Ex|x>53 -2 -47-7 X >7 4x-1bZ0 4I6 +1b
Nxz!6 x y X>t
31=x33x+3 31 -31 O=x*-3x-28 (x-)(x+) T
Form combinations (sum, difference, product and quotient) of two functions:
Sum:
(+)()=()+()
Difference:
(2()=(2() ()()=():() (-)()=X()+0 ()
Product:
Quotient:
(-00-2)u(-22)u (2,00) Note: The domain of the combined function is the union of each individual function9s (f and g) domain, except for the quotient which must exclude values of x that cause division by zero (in other words,() # 0)
Example7: Find the v