<< Chapter < Page Chapter >> Page >

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.

Got questions? Get instant answers now!

Algebraic

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

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

Got questions? Get instant answers now!

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

function

Got questions? Get instant answers now!

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

y = 2 x 2 + 40 x

function

Got questions? Get instant answers now!

x = 3 y + 5 7 y 1

function

Got questions? Get instant answers now!

y = 3 x + 5 7 x 1

function

Got questions? Get instant answers now!

y 2 = x 2

not a function

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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 |

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

  1. Evaluate k ( 2 ) .
  2. Solve k ( t ) = 7.
Got questions? Get instant answers now!

Given the function f ( x ) = 8 3 x :

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

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

Got questions? Get instant answers now!

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

  1. Evaluate p ( −3 ) .
  2. Solve p ( c ) = 2.
Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

  1. Evaluate f ( 7 ) .
  2. Solve f ( x ) = 4.
Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Given the following graph,

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

Graph of relation.
Got questions? Get instant answers now!

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

Got questions? Get instant answers now!
Graph of a one-to-one function.

function, but not one-to-one

Got questions? Get instant answers now!

Numeric

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

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

Got questions? Get instant answers now!

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

function

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!
x 5 10 10
y 3 8 14

not a function

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

f ( x ) = 3 + x + 3

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

Got questions? Get instant answers now!

f ( x ) = 3 x

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

Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!

f ( 7 3 ) h ( 2 )

20

Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

[ 100 ,  100 ]

[ 1 , 000 , 000 ,  1,000,000 ]

Graph of a cubic function.
Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

[ −1,000,000 , 1,000,000 ]

[ 100 ,  100 ]

Graph of a cubic root function.
Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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
Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Questions & Answers

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
Please you teach
Dua
Yes
ahmed
Thank you
Dua
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask