10.1 function and relations
TRANSCRIPT
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10.1 Functions andRelations
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Relation Let X and Y denote sets of real
numbers and let (x, y) denote an
ordered pair of numbers such thatfor each x X there is at least one
y Y. Then a set of number pairs
satisfying this condition is called arelation.
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Domain and Range The set of all the first elements of
the ordered pairs is called the
domain of the relation, and the setof all the second elements is calledthe range.
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Function A relation in which every ordered
pair (x, y) has one and only one
value of y corresponding to thevalue of x is called a function.
A function is a relation in which no
two ordered pairs have the samefirst element and different secondelements.
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List the elements of the domain and theelements of the range of each set of
ordered pairs. Tell which are functions.
1. {(2, 1), (4, -3), (3, -1)}
2. {(2, 4), (0, 5), (1, -3)}
3. {(2, 0), (2, 3), (5, 0)}
4. {(5, 1), (1, 5), (5, 2)}
5. {(0, -3), (1, -2), (0, 1)}6. {(-1, 0), (0, 1), (0, 0)}
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Equations and Graphical
Representations of Relations1. y = mx + b
Line with slope m
and y-intercept b.
Function
Domain: x
Range: y
y
x
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2. (x-h)
2
+ (y-k)
2
= r
2
Circle with center (h,k)
and radius r
Relation
Domain: h-r x h+r
Range: k-r
y
k+r
r (h,k)
y
x
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3. (y-k) = √r2
– (x-h)
2
Semicircle with center
(h,k) and radius r
Function
Domain: h-r x h+r
Range: k
y
k+r
r
(h, k)
y
x
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(x-h) = √r
2
– (y-k)
2
Semicircle with center
(h,k) and radius r
Relation
Domain: h x h+r
Range: k-r
y
k+r
r
(h,k)
y
x
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4. y = a(x-h)
2
+ k Parabola with vertex (h, k)
axis of symmetry: line x=h
Opens up if a > 0Opens down if a < 0
Function
Domain: x
Range: y k if parabola opens up
y k if parabola opens down
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x = a(y-k)
2
+ hParabola with vertex (h,k)
Axis of Symmetry: line y=k
Opens to the right if a > 0Opens to the left if a < 0
Relation
Domain: x h if parabola opens to the right
x h if parabola opens to the left
Range: y
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5. y = √a(x-h) + k Half parabola
Function
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x = √a(y-k) + hHalf parabola
Function
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6. y = alx-hl + k V-shaped graph
Vertex: (h, k)
Function
Domain: x
Range: y
k if the graph opens up y k if the graph opensdown
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y = aly-kl + h V-shaped graph
Relation
Vertex (h, k)
Domain: x h if the graph opens to theright
x h if the graph opens to theleft
Range: y
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7. x2-y2 = 1a2 b2
Hyperbola
Relation
Domain: x
Range: y a or y -a
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y2 – x2 = 1a2 b2
Hyperbola
Relation
Domain: x a or x -a
Range: y
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8. x2 + y2 = 1a2 b2
Ellipse
Relation
Domain: -a x a
Range: -b y b
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9. Graphs with asymptotes
y
x
Domain: x 0
Range: y 0
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10. Inequalities
Relation
y < mx + b
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Vertical Line Test
If a vertical line intersects a graphin exactly one point, then the
graph represents a function.
Function Not a Function
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Exercises
A. Which of the following setsdescribe a function, and which
describe only a relation?1. {(1, 2), (2, 3), (3, 4)}
2. {(1, 2), (1, 3), (2, 4)}
3. {(x, y)l y = 2x + 4}
4. {(x, y)l 2x + 3y = 6}
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5. {(x, y)l x + y2 = 1}
6. {(x, y)l y + x2 = 1}
7. {(x, y)l (x-2)2 + (y-3)2 = 9
8. {(x, y)l x y}
9. {(x, y)l y + 1 = 7 x - 3
10. {(x, y)l lxl + lyl = 1
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B. What is the domain and range of the function and relation in #1-10?
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C. Give the domain and range of thefollowing functions, where x, y .
1. y = x 7. y = 1
2. y = √x (x2 – 1)
3. y = x2 8. y = lxl - 2
4. y = √4 – x2
5. y = √x2 – 1
6. y = x
(1-x)
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D. If f(x) = 2x – 3, find
1. f(0) 3. f(3)
2. f(1) 4. f(-4)
E. If f(x) = x2 -7x + 10, find
1. f(4) 3. f(3)2. f(5) 4. f(0)
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F. If f(x) = 1/(x – 3) , find
1. f(0) 3. f(-1)
2. f(2) 4. f(3)
G. If e(x) = 2 x , find
1. e(0) 3. e(5)2. e(1) 4. e(-5)