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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 , evaluate g ( x + h ) g ( x ) h , h 0.

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Given the function g ( x ) = x 2 + 2 x , evaluate 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

difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
Feemark Reply
can you not take the square root of a negative number
Sharon Reply
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
Elaine Reply
can I get some pretty basic questions
Ama Reply
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
Amara Reply
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
Mars Reply
what is the domain of f(x)=x-4/x^2-2x-15 then
Conney Reply
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
jeric Reply
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
jeric Reply
What are the question marks for?
Elliott

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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