Mrs.Volynskaya Relations Domain Range A relation is a mapping, or pairing, of input values with output values. The set of input values is called the domain. The set of output values is called the range.
Domain is the set of all x values. Domain & Range Domain is the set of all x values. Range is the set of all y values. {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)} Example 1: Domain- Range- D: {1, 2} R: {1, 2, 3}
A Relation can be represented by a set of ordered pairs of the form (x,y) Quadrant II X<0, y>0 Quadrant I X>0, y>0 Origin (0,0) Quadrant IV X>0, y<0 Quadrant III X<0, y<0
Plot: (-3,5) (-4,-2) (4,3) (3,-4)
Most equations have infinitely many solution points. Every equation has solution points (points which satisfy the equation). 3x + y = 5 (0, 5), (1, 2), (2, -1), (3, -4) Some solution points: Most equations have infinitely many solution points. Page 111
Ex 5. Graph y = x² - 5 x x² - 5 y -3 -2 -1 1 2 3
Functions In order for a relationship to be a function… EVERY INPUT MUST HAVE AN OUTPUT TWO DIFFERENT INPUTS CAN HAVE THE SAME OUTPUT ONE INPUT CAN HAVE ONLY ONE OUTPUT INPUT Functions (DOMAIN) FUNCTION MACHINE OUTPUT (RANGE)
Identify the Domain and Range. Then tell if the relation is a function. Input Output -3 3 1 1 3 -2 4 Function? Yes: each input is mapped onto exactly one output Domain = {-3, 1,3,4} Range = {3,1,-2}
Identify the Domain and Range. Then tell if the relation is a function. Input Output -3 3 1 -2 4 1 4 Domain = {-3, 1,4} Range = {3,-2,1,4} Notice the set notation!!! Function? No: input 1 is mapped onto Both -2 & 1
The Vertical Line Test If it is possible for a vertical line to intersect a graph at more than one point, then the graph is NOT the graph of a function. Page 117
Use the vertical line test to visually check if the relation is a function. (4,4) (-3,3) (1,1) (1,-2) Function? No, Two points are on The same vertical line.
Use the vertical line test to visually check if the relation is a function. (-3,3) (1,1) (3,1) (4,-2) Function? Yes, no two points are on the same vertical line
YES! Function? #1
#2 Function? YES! Y = 0.5x + 2 + 2sin(x) D: all reals R: all reals Another cool function: abs(x) + 2sin(x)
#3 Function? NO! Y = 0.5x + 2 + 2sin(x) D: all reals R: all reals Another cool function: abs(x) + 2sin(x)
#4 Function? YES! Y = 0.5x + 2 + 2sin(x) D: all reals R: all reals Another cool function: abs(x) + 2sin(x)
#5 Function? NO!
YES! Function? #6 This is a piecewise function
Function? #7 NO! D: all reals R: [0, 1] Another cool function: y = sin(abs(x)) Y = sin(x) * abs(x)
#8 Function? NO! Y = 0.5x + 2 + 2sin(x) D: all reals R: all reals Another cool function: abs(x) + 2sin(x)
YES! #9 Function?
Function? #10 YES!
Function? #11 NO! D: [-3, -1) U (-1, 3] R: {-1, 1}
YES! Function? #12 D: [-3, -1) U (-1, 3] R: {-1, 1}
Function Notation “f of x” Input = x Output = f(x) = y
(x, y) (x, f(x)) (input, output) y = 6 – 3x f(x) = 6 – 3x x y x f(x) Before… Now… y = 6 – 3x f(x) = 6 – 3x x y x f(x) -2 -1 1 2 12 -2 -1 1 2 12 (x, y) (x, f(x)) 9 9 6 6 3 3 (input, output)
g(2) = 2 3 g(5) = Find g(2) and g(5). Example 7 Find g(2) and g(5). g = {(1, 4),(2,3),(3,2),(4,-8),(5,2)} g(2) = 2 3 g(5) =
Example 9. f(x) = 2x2 – 3 Find f(0), f(-3), f(5a).
F(x) = 3x2 +1 Find f(0), f(-1), f(2a). f(0) = 1 f(-1) = 4 Example 10. F(x) = 3x2 +1 Find f(0), f(-1), f(2a). f(0) = 1 f(-1) = 4 f(2a) = 12a2 + 1
D: All real numbers except -3 What is the domain? g(x) = -3x2 + 4x + 5 Ex. D: all real numbers x + 3 0 Ex. x -3 D: All real numbers except -3
What is the domain? h x ( ) = - 1 5 f x ( ) = + 1 2 x - 5 0 x + 2 0 Ex. D: All real numbers except 5 Ex. f x ( ) = + 1 2 x + 2 0 D: All Real Numbers except -2
What are your questions?