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Why does the horizontal line test tell us whether the graph of a function is one-to-one?

When a horizontal line intersects the graph of a function more than once, that indicates that for that output there is more than one input. A function is one-to-one if each output corresponds to only one input.

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Algebraic

For the following exercises, determine whether the relation represents a function.

{ ( a , b ) ,   ( c , d ) ,   ( a , c ) }

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{ ( a , b ) , ( b , c ) , ( c , c ) }

function

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For the following exercises, determine whether the relation represents y as a function of x .

y = 2 x 2 + 40 x

function

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x = 3 y + 5 7 y 1

function

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y = 3 x + 5 7 x 1

function

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y 2 = x 2

not a function

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For the following exercises, evaluate the function f at the indicated values   f ( −3 ) , f ( 2 ) , f ( a ) , f ( a ) , f ( a + h ) .

f ( x ) = 2 x 5

f ( 3 ) = 11 ; f ( 2 ) = 1 ; f ( a ) = 2 a 5 ; f ( a ) = 2 a + 5 ; f ( a + h ) = 2 a + 2 h 5

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f ( x ) = 5 x 2 + 2 x 1

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f ( x ) = 2 x + 5

f ( 3 ) = 5 + 5 ; f ( 2 ) = 5 ; f ( a ) = 2 + a + 5 ; f ( a ) = 2 a 5 ; f ( a + h ) = 2 a h + 5

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f ( x ) = 6 x 1 5 x + 2

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f ( x ) = | x 1 | | x + 1 |

f ( 3 ) = 2 ; f ( 2 ) = 1 3 = 2 ; f ( a ) = | a 1 | | a + 1 | ; f ( a ) = | a 1 | + | a + 1 | ;   f ( a + h ) = | a + h 1 | | a + h + 1 |

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Given the function g ( x ) = 5 x 2 , simplify g ( x + h ) g ( x ) h , h 0.

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Given the function g ( x ) = x 2 + 2 x , simplify g ( x ) g ( a ) x a , x a .

g ( x ) g ( a ) x a = x + a + 2 , x a

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Given the function k ( t ) = 2 t 1 :

  1. Evaluate k ( 2 ) .
  2. Solve k ( t ) = 7.
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Given the function f ( x ) = 8 3 x :

  1. Evaluate f ( 2 ) .
  2. Solve f ( x ) = −1.

a. f ( 2 ) = 14 ; b. x = 3

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Given the function p ( c ) = c 2 + c :

  1. Evaluate p ( −3 ) .
  2. Solve p ( c ) = 2.
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Given the function f ( x ) = x 2 3 x :

  1. Evaluate f ( 5 ) .
  2. Solve f ( x ) = 4.

a. f ( 5 ) = 10 ; b. x = 1   or   x = 4

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Given the function f ( x ) = x + 2 :

  1. Evaluate f ( 7 ) .
  2. Solve f ( x ) = 4.
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Consider the relationship 3 r + 2 t = 18.

  1. Write the relationship as a function r = f ( t ) .
  2. Evaluate f ( −3 ) .
  3. Solve f ( t ) = 2.

a. f ( t ) = 6 2 3 t ; b. f ( 3 ) = 8 ; c. t = 6

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Graphical

For the following exercises, use the vertical line test to determine which graphs show relations that are functions.

Given the following graph,

  • Evaluate f ( −1 ) .
  • Solve for f ( x ) = 3.

Graph of relation.
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Given the following graph,

  • Evaluate f ( 0 ) .
  • Solve for f ( x ) = −3.

Graph of relation.

a. f ( 0 ) = 1 ; b. f ( x ) = 3 , x = 2   or   x = 2

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Given the following graph,

  • Evaluate f ( 4 ) .
  • Solve for f ( x ) = 1.

Graph of relation.
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For the following exercises, determine if the given graph is a one-to-one function.

Graph of a circle.

not a function so it is also not a one-to-one function

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Graph of a one-to-one function.

function, but not one-to-one

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Numeric

For the following exercises, determine whether the relation represents a function.

{ ( −1 , −1 ) , ( −2 , −2 ) , ( −3 , −3 ) }

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{ ( 3 , 4 ) , ( 4 , 5 ) , ( 5 , 6 ) }

function

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{ ( 2 , 5 ) , ( 7 , 11 ) , ( 15 , 8 ) , ( 7 , 9 ) }

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For the following exercises, determine if the relation represented in table form represents y as a function of x .

x 5 10 15
y 3 8 14

function

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x 5 10 10
y 3 8 14

not a function

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For the following exercises, use the function f represented in [link] .

x f ( x )
0 74
1 28
2 1
3 53
4 56
5 3
6 36
7 45
8 14
9 47

Solve f ( x ) = 1.

f ( x ) = 1 , x = 2

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For the following exercises, evaluate the function f at the values f ( 2 ) , f ( −1 ) , f ( 0 ) , f ( 1 ) , and f ( 2 ) .

f ( x ) = 8 3 x

f ( 2 ) = 14 ; f ( 1 ) = 11 ; f ( 0 ) = 8 ; f ( 1 ) = 5 ; f ( 2 ) = 2

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f ( x ) = 8 x 2 7 x + 3

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f ( x ) = 3 + x + 3

f ( 2 ) = 4 ;    f ( 1 ) = 4.414 ; f ( 0 ) = 4.732 ; f ( 1 ) = 4.5 ; f ( 2 ) = 5.236

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f ( x ) = 3 x

f ( 2 ) = 1 9 ; f ( 1 ) = 1 3 ; f ( 0 ) = 1 ; f ( 1 ) = 3 ; f ( 2 ) = 9

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For the following exercises, evaluate the expressions, given functions f , g , and h :

  • f ( x ) = 3 x 2
  • g ( x ) = 5 x 2
  • h ( x ) = −2 x 2 + 3 x 1

3 f ( 1 ) 4 g ( 2 )

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f ( 7 3 ) h ( 2 )

20

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Technology

For the following exercises, graph y = x 2 on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.

[ 10 ,  10 ]

[ 0 ,  100 ]

Graph of a parabola.
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For the following exercises, graph y = x 3 on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.

[ 0.1 ,  0 .1 ]

[ 0.001 ,  0 .001 ]

Graph of a parabola.
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[ 100 ,  100 ]

[ 1 , 000 , 000 ,  1,000,000 ]

Graph of a cubic function.
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For the following exercises, graph y = x on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.

[ 0 ,  100 ]

[ 0 ,  10 ]

Graph of a square root function.
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For the following exercises, graph y = x 3 on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.

[ −0.001 , 0.001 ]

[ −0.1 , 0.1 ]

Graph of a square root function.
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[ −1,000,000 , 1,000,000 ]

[ 100 ,  100 ]

Graph of a cubic root function.
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Real-world applications

The amount of garbage, G , produced by a city with population p is given by G = f ( p ) . G is measured in tons per week, and p is measured in thousands of people.

  1. The town of Tola has a population of 40,000 and produces 13 tons of garbage each week. Express this information in terms of the function f .
  2. Explain the meaning of the statement f ( 5 ) = 2.
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The number of cubic yards of dirt, D , needed to cover a garden with area a square feet is given by D = g ( a ) .

  1. A garden with area 5000 ft 2 requires 50 yd 3 of dirt. Express this information in terms of the function g .
  2. Explain the meaning of the statement g ( 100 ) = 1.

a. g ( 5000 ) = 50 ; b. The number of cubic yards of dirt required for a garden of 100 square feet is 1.

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Let f ( t ) be the number of ducks in a lake t years after 1990. Explain the meaning of each statement:

  1. f ( 5 ) = 30
  2. f ( 10 ) = 40
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Let h ( t ) be the height above ground, in feet, of a rocket t seconds after launching. Explain the meaning of each statement:

  1. h ( 1 ) = 200
  2. h ( 2 ) = 350

a. The height of a rocket above ground after 1 second is 200 ft. b. the height of a rocket above ground after 2 seconds is 350 ft.

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Show that the function f ( x ) = 3 ( x 5 ) 2 + 7 is not one-to-one.

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Questions & Answers

(x2-2x+8)-4(x2-3x+5)
Ayush Reply
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
Y
master
X2-2X+8-4X2+12X-20=0 (X2-4X2)+(-2X+12X)+(-20+8)= 0 -3X2+10X-12=0 3X2-10X+12=0 Use quadratic formula To find the answer answer (5±Root11i)/3
master
Soo sorry (5±Root11* i)/3
master
x2-2x+8-4x2+12x-20 x2-4x2-2x+12x+8-20 -3x2+10x-12 now you can find the answer using quadratic
Mukhtar
explain and give four example of hyperbolic function
Lukman Reply
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
Racelle Reply
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Anurag Reply
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
Jhovie Reply
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
thanks
Jhovie
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
Charmaine Reply
A banana.
Yaona
given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
Snalo Reply
what are you up to?
Mark Reply
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
sita Reply
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
Pearl Reply
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
haha. already finished college
Jeffrey
how about you? what grade are you now?
Jeffrey
I'm going to 11grade
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
Nelson Reply
which equation
Miranda
I dont know. lol
Jeffrey
please where is the equation
Miranda
ask nelson. lol
Jeffrey
answer and questions in exercise 11.2 sums
Yp Reply
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
Swadesh
cos(- z)=cos z
Mustafa
what is a algebra
Jallah Reply
(x+x)3=?
Narad
6x
Obed
what is the identity of 1-cos²5x equal to?
liyemaikhaya Reply
__john __05
Kishu
Hi
Abdel
hi
Ye
hi
Nokwanda
C'est comment
Abdel
Hi
Amanda
hello
SORIE
Hiiii
Chinni
hello
Ranjay
hi
ANSHU
hiiii
Chinni
h r u friends
Chinni
yes
Hassan
so is their any Genius in mathematics here let chat guys and get to know each other's
SORIE
I speak French
Abdel
okay no problem since we gather here and get to know each other
SORIE
hi im stupid at math and just wanna join here
Yaona
lol nahhh none of us here are stupid it's just that we have Fast, Medium, and slow learner bro but we all going to work things out together
SORIE
it's 12

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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