10.1 function and relations

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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)