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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

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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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