01 - functions and their graphs - part 1 - copy

8/2/2019 01 - Functions and Their Graphs - Part 1 - Copy

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Relation a set of ordered pairs, which containsthe pairs of abscissa and ordinate. The firstnumber in each ordered pair is the x-value or the

abscissa, and the second number in eachordered pair is the y-value, or the ordinate.

Domain is the set of all the abscissas, and

range is the set of all ordinates.


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A relation may also be shown using a table of values orthrough the use of a mapping diagram.

Illustration:

Using a table. Using a mapping diagram.

Domain Range0 11 22 33 44 57 8


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Function , denoted by f, is a rule that assignsto each element x in a set X exactly one elementf(x) in a set Y.

The set X is called the domain of the functionand Y its codomain.

The set of assigned elements in Y is called therange of the function f.

The function notation f(x) means the value offunction fusing the independent number x.


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Given the ordered pairs below, determine if it isa mere relation or a function.

(0,1) , (1, 2), (2, 3), (3, 4), (4, 5), (7, 8)

Answer:

For every given x-value there is acorresponding unique y-value. Therefore, therelation is a function.


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Which relation represents a function?

A. {(1,3), (2, 4), (3,5), (5, 1)}

B. {(1, 0), (0,1), (1, -1)}

C. {(2, 3), (3, 2), (4, 5), (3, 7)}

D. {(0, 0), (0, 2)}

Answer:

A


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Which mapping diagram does not represent a function?

A. B.

C. D.


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If f(x) = x2 + 3x+ 5, evaluate:

a. f(2) b. f(x+ 3) c. f(-x)

Solutiona. We find f(2) by substituting 2 for x in theequation.

f(2) = 22+ 3 2 + 5 = 4 + 6 + 5 = 15

Thus, f(2) = 15.


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Solution

b. We find f(x+ 3) by substituting (x+ 3) for x inthe equation.f(x+3) = (x+3)2 + 3(x+3)+ 5

If f(x) = x2 + 3x+ 5, evaluate: b. f(x+ 3)

Equivalently,

f(x+3) = (x+3)2 + 3(x+3)+ 5= x2 + 6x+ 9 + 3x+ 9 + 5= x2 + 9x+ 23.


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Solution

c. We find f(-x) by substituting (-x)for x in theequation.

f(-x) = (-x)2 + 3(-x)+ 5

If f(x) = x2 + 3x+ 5, evaluate: c. f(-x)

Equivalently,

f(-x) = (-x)2 + 3(-x)+ 5= x23x+ 5.


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Which is the range of the relation describedby y = 3x 8 if its domain is {-1, 0, 1}?

A) {-11, 8, 5}

B) {-5, 0 5}

C) {-11, -8, -5}

D) {0, 3, 5}


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Which is the range of the relation describedby 3y = 2x2 36 if its domain is {3, 6, 9}?

A) {-6, 12, 42}

B) {6, 12, 42}

C) {0, 6, 12}

D) {-6, 0, 12}


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Find the domain and range of each function.

1. 2.

1.

Domain:

Range:

2.

Domain: R except 2

Range: R except 0

4)( xxf2

2)(

x

xf

4x

0y


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Let f and g be two functions. The sum, thedifference, the product , and the quotient arefunctions whose domains are the set of all real

numbers common to the domains of fand g,defined as follows:

Sum: (f+ g)(x) = f (x)+g(x)

Difference: (fg)(x) = f (x) g(x)

Product: (f g)(x) = f (x) g(x)

Quotient: (f/g)(x) = f (x)/g(x), g(x) 0


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Let f(x) = 2x+1 and g(x) = x2 - 2.

Find

a. (f + g) (x) c.(g f) (x) e. (f / g) (x)

b. (f g) (x) d. (f g) (x) f. (g/f) (x)Solution:

a. (f + g) (x) = f(x) + g( x) = (2x+1 )+ (x2 2) = x2 + 2x- 1

b. (f g)(x) = f(x) - g(x) = (2x+1) - (x2 - 2) = -x2 + 2x +3

c. (g f)(x) = g(x) - f(x) = (x2 - 2) (2x +1) = x2 - 2x - 3

d. (f g)(x) = f(x) g(x) = (2x+1)(x2

- 2) = 2x3

+ x2

- 4x -2

2x

2

1

x


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Let f(x) = 3x+6 and g(x) = x +2.Finda. (f + g) (1)b. (f g) (2)

c. (f g) (0)d. (f/g) (-1)e. (g/f) (-1)ANSWERS:

a. (f + g) (1) = 12b. (f g) (2) = 8c. (f g) (0) = 12d. (f/g) (-1) = 3

e. (g/f) (-1) = 1/3


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The composition of the function fwith gisdenoted by fogand is defined by the equation

(fog)(x) = f(g(x)).

The domain of the composite function fogis

the set of all xsuch that xis in the domain of gand g(x) is in the domain of f.


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Given f(x) = 3x 4 and g(x) = x2 + 6,

find: a. (fg)(x) b. (gf)(x)Solutiona. We begin with (fog)(x), the composition of fwith g. Because (fo

g)(x) means f(g(x)), we must replace each occurrence of xin theequation for f by g(x).

f(x) = 3x 4(fg)(x) = f(g(x)) = 3(g(x)) 4

= 3(x2 + 6) 4= 3x2 + 18 4= 3x2 + 14

Thus, (fg)(x) = 3x2 + 14.


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Solutionb. Next, we find (gof)(x), the composition of gwith f.

Because (gof)(x) means g(f(x)), we must replace eachoccurrence of xin the equation for g by f(x).

g(x) = x2 + 6

(gf)(x) = g(f(x)) = (f(x))2 + 6= (3x 4)2 + 6= 9x2 24x+ 16 + 6= 9x2 24x+ 22

Notice that (fg)(x) is not the same as (gf)(x).

Given f(x) = 3x 4 and g(x) = x2 + 6,find: a. (fg)(x) b. (gf)(x)


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Given f(x) = x 2 and g(x) = x+ 7,find: a. (fg)(x)

b. (g f)(x)

c. (ff)(x)d. (gg)(x)

Answers:a. (fg)(x) = x + 5b. (g f)(x) = x + 5

c. (ff)(x) = x - 4

d. (gg)(x) = x + 14


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If fis a function, then the graph of fis the set of all points(x,y) in the Cartesian plane for which (x,y) is an orderedpair in f.

The graph of a function can be intersected by a verticalline in at most one point.

Vertical Line Test

o If a vertical line intersects a graph more than once, then thegraph is not the graph of a function.


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Determine if the graph is a graph of a function or just agraph of a relation.

8

6

4

2

-2

-4

5 10 15

graph

of arelation


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Determine if thegraph is a graph of afunction or just a

graph of a relation.

graph

of afunction


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Determine if thegraph is a graph ofa function or just a


graph

of arelation


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16

14

12

10

8

6

2

2

6

8

15 10 5 5 10 15 20 25

A

Determine if thegraph is a graph of afunction or just a


graph

of arelation


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3

2

1

-1

-2

-3

-4

- 6 - 4 -2 2 6

graphof a

relation


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6

4

2

-2

-4

-6

10 -5 5 10


graphof a

relation


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graphof a

function

3

1

-3 -2 -1 1 2 3 4

-1

-2

-3

-5


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Domain: The set of real

numbers

(-, ) x

Range:

The set of realnumbers

(-, ) y


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10

9

8

7

6

5

4

3

1

-1

-2

-8 -6 -4 -2 4 6 8


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numbers

(-, ) x

Range:

[-1, ) y -1


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16

14

12

10

8

6

4

2

-20 -15 -10 -5 5 10


numbers

(-, ) x

Range: [0, ) y 0


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Domain: (-, 0) (0, ) The set of real

numbers, except 0

Range:

(-, -1) (1, ) The set of real

numbers except[-1,1]


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Constantf(x1) < f(x2)

(x1, f(x1))

(x2, f(x2))

Increasingf(x1) < f(x2)

(x1, f(x1))

(x2, f(x2))

Decreasingf(x1) < f(x2)

(x1, f(x1))

(x2, f(x2))

A function is increasing on an interval if for any x1, and x2 in theinterval, where x1 < x2, then f(x1) < f(x2).A function is decreasing on an interval if for any x1, and x2 in theinterval, where x1 < x2, then f(x1) > f(x2).A function is constant on an interval if for any x

1

, and x2

in theinterval, where x1 < x2, then f(x1) = f(x2).


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Solutiona. The function is decreasing on the interval (-, 0), increasing on the

interval (0, 2), and decreasing on the interval (2, ).

Describe the increasing, decreasing, or constant behavior of eachfunction whose graph is shown.

-5 -4 -3 -2 -1 1 2 3 4 5

5

4

3

1

-1

-2

-3

-4

-5

-5 -4 -3 -2 -1 1 2 3 4 5

5

4

3

2

1

-1

-2

-3

-4

-5

a.

b.


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Solution:b.

Although the function's equations are not given, the graph indicatesthat the function is defined in two pieces.

The part of the graph to the left of the y-axis shows that thefunction is constant on the interval (-, 0).

The part to the right of the y-axis shows that the function is

increasing on the interval [0,).

Describe the increasing, decreasing, or constant behavior of each functionwhose graph is shown.

-5 -4 -3 -2 -1 1 2 3 4 5

5

4

3

1

-1

-2

-3

-4

-5

-5 -4 -3 -2 -1 1 2 3 4 5

5

4

3

2

1

-1

-2

-3

-4

-5

a.

b.


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Describe theincreasing,decreasing, or

constant behavior ofeach function whosegraph is shown.

Decreasing on (-, 0);

Increasing on (0, )


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Increasing on (-, 2);

Constant on (2, )


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Increasing on (-,)


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A continuous function is represented by a graphwhich may be drawn using a continuous line or curve,while a discontinuous function is represented by agraph which has some gaps, holes or breaks

(discontinuities).


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A periodic function is a function whose values repeatin periods or regular intervals.

y = tan(x) y = cos(x)


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A linear function is a function of the form f(x) = mx +bwhere mand bare real numbers and m 0.Domain: the set of real numbersRange: the set of real numbers

Graph: straight lineExample: f(x) = 2 - x


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General Form ax + by + c = 0 wherein a, b, and care real numbers

Slope-InterceptForm y = mx + b wherein m is theslope of the line, andb is the y-intercept

Point-SlopeForm y y1 = m (xx1)

wherein m is theslope of the line, andP(x1, y1) is a point onthe line


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Find an equation for the line through (-2, 5) andslope -3.

Solution:

11 xxmyy

)2(35 xy

635 xy

13 xy


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Find the equation of the line through the givenpair of points (3,5) and (4,7).

Solution:

Find the slopeUse the slope and one point, say (3, 5) in the

point-slope form

234

57

mslope

11 xxmyy

325 xy

625 xy

12 xy


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1. Find the slope of the line that passes through (2,3) and(4,3).

2. Find the slope and the y-intercept of 3x + 5y - 9 = 0.

3. What is the slope of the line y - 4x + 6 = 0?4. What is the y-intercept of the line 3x + 2y = 5?

5. What is an equation of the line through (4,1) and (2,4) ?

6. What is an equation of the line passing through the

points ( 6, -3 ) and ( -2, 3 )?


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7. Find an equation of the line which has a slope of 2/3and a y-intercept of 2?

8. Find an equation of the line with x-intercept -2 and y-intercept 2?

9. What is an equation of the line through (-8,1) withundefined slope ?

10. What is an equation of the line through (4,3) with slopeequal to zero?


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For each of the given equations, do the following:

Rewrite the equation in slope-intercept form

Determine the slope.

Find the intercepts (x and y). Graph the equation.

1. y 5x 10 = 0

2. 2y x + 4 = 0

3. -2x + y + 8 = 04. 3y + 2x + 6 = 0

5. 5x 5y 15 = 0


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A quadratic function is a function of the formf(x) = ax2+bx +cwhere a, band c are real numbers and a 0.Domain: the set of real numbersGraph: parabola

Examples: parabolas parabolasopening upward opening downward


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The graph of any quadratic function is called a parabola.Parabolas are shaped like cups, as shown in the graphbelow.

If the coefficient of x2 is positive, the parabola opens

upward; otherwise, the parabola opens downward.

The vertex (or turning point) is the minimum or maximumpoint.

Graphs of Quadratic Functions


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Given f(x) = ax2+ bx +c1. Determine whether the parabola opens upward or

downward. If a>0, it opens upward. If a< 0, it opensdownward.

2. Determine the vertex of the parabola. The vertex is

3. The axis of symmetry is

The axis of symmetry divides the parabola into two equal

parts such that one part is a mirror image of the other.

a

bac

a

b

4

4,

2

2

a

bx

2


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Given f(x) = ax2+ bx +c

4. Find any x-intercepts by replacing f (x) with 0. Solve theresulting quadratic equation for x.

5. Find the y-intercept by replacing x with zero.

6. Plot the intercepts and vertex. Connect these points witha smooth curve that is shaped like a cup.


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The function f(x) = 1 - 4x - x2 has its vertexat _____.

A. (2,11) B. (2,-11)

C.( -2,-3)

D.(-2,5)


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Identify the graph of the given function: y = 3x2 - 3.


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Identify the graph of the given function: 4y = x2.


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Identify the graph of the given function: y = (x - 2)(x 2).


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Consider f(x) = ax2 + bx +c.

1. If a > 0, then fhas a minimum that occurs atx = -b/(2a). This minimum value is f(-b/(2a)).

2. If a < 0, the fhas a maximum that occurs atx = -b/(2a). This maximum value is f(-b/(2a)).


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The maximum value of the functionf(x) = -3x2 2x + 4 is ____.

A. 13/3

B. 3/13

C. 9

D. 13


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The function f(x) = x2 8x + 16 has _____.

A. a minimum value at x = -4

B. a maximum value at x = -4

C. a minimum value at x = 4

D. a maximum value at x = 4


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An absolute value function fis defined by

Domain: the set of real numbersGraph: v-shapedExamples: y = -|x| y = |x| y = x - |x|


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A Rational Function is a function in the form:

where p(x) and q(x) are polynomial functions and q(x) 0.

Examples:

)(

)()(

xq

xpxf

xy

24

x

xy

8 Polynomial Functions


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A polynomial function is a function of the form:

on

n

n

naxaxaxaxf

1

1

1

All of these coefficients are real numbers

nmust be a positive integer

The degree of the polynomial is the largestpower on any xterm in the polynomial.

an 0

8. Polynomial Functions


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y x3 4x

y x3 2x2 x 4


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A piecewise-defined function (or piecewise function) isa function whose definition changes depending on the valueof the independent variable (x).A real-valued function f of a real variable x is a relationship

whose definition is given differently on disjoint subsets of itsdomain ( called subdomains).The word piecewise is also used to describe any property ofa piecewise-defined function that holds for each piece but

may not hold for the whole domain of the function.

Examples:

1,2

1,32)(

x

xxxf


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1. 5.

2.

3. 6.

4. 7.

>3,3,32)(

xxxxxf

>

1,3

1,3)(

xx

xxxf

0,1

0,2)(

x

xxxf

3,2

3,3)(

x

xxxf

>


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http://rechneronline.de/function-graphs/

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01 - functions and their graphs - part 1 - copy

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