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The most common graphs name the input value x and the output value y , and we say y is a function of x , or y = f ( x ) when the function is named f . The graph of the function is the set of all points ( x , y ) in the plane that satisfies the equation y = f ( x ) . If the function is defined for only a few input values, then the graph of the function consists of only a few points, where the x -coordinate of each point is an input value and the y -coordinate of each point is the corresponding output value. For example, the black dots on the graph in [link] tell us that f ( 0 ) = 2 and f ( 6 ) = 1. However, the set of all points ( x , y ) satisfying y = f ( x ) is a curve. The curve shown includes ( 0 , 2 ) and ( 6 , 1 ) because the curve passes through those points.

Graph of a polynomial.

The vertical line test    can be used to determine whether a graph represents a function. If we can draw any vertical line that intersects a graph more than once, then the graph does not define a function because a function has only one output value for each input value. See [link] .

Three graphs visually showing what is and is not a function.

Given a graph, use the vertical line test to determine if the graph represents a function.

  1. Inspect the graph to see if any vertical line drawn would intersect the curve more than once.
  2. If there is any such line, determine that the graph does not represent a function.

Applying the vertical line test

Which of the graphs in [link] represent(s) a function y = f ( x ) ?

Graph of a polynomial.

If any vertical line intersects a graph more than once, the relation represented by the graph is not a function. Notice that any vertical line would pass through only one point of the two graphs shown in parts (a) and (b) of [link] . From this we can conclude that these two graphs represent functions. The third graph does not represent a function because, at most x -values, a vertical line would intersect the graph at more than one point, as shown in [link] .

Graph of a circle.
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Does the graph in [link] represent a function?

Graph of absolute value function.

yes

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Using the horizontal line test

Once we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the horizontal line test    . Draw horizontal lines through the graph. If any horizontal line intersects the graph more than once, then the graph does not represent a one-to-one function.

Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function.

  1. Inspect the graph to see if any horizontal line drawn would intersect the curve more than once.
  2. If there is any such line, determine that the function is not one-to-one.

Applying the horizontal line test

Consider the functions shown in [link] (a) and [link] (b) . Are either of the functions one-to-one?

The function in [link] (a) is not one-to-one. The horizontal line shown in [link] intersects the graph of the function at two points (and we can even find horizontal lines that intersect it at three points.)

The function in [link] (b) is one-to-one. Any horizontal line will intersect a diagonal line at most once.

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Is the graph shown in [link] one-to-one?

No, because it does not pass the horizontal line test.

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Identifying basic toolkit functions

In this text, we will be exploring functions—the shapes of their graphs, their unique characteristics, their algebraic formulas, and how to solve problems with them. When learning to read, we start with the alphabet. When learning to do arithmetic, we start with numbers. When working with functions, it is similarly helpful to have a base set of building-block elements. We call these our “toolkit functions,” which form a set of basic named functions for which we know the graph, formula, and special properties. Some of these functions are programmed to individual buttons on many calculators. For these definitions we will use x as the input variable and y = f ( x ) as the output variable.

Questions & Answers

sin^4+sin^2=1, prove that tan^2-tan^4+1=0
SAYANTANI Reply
what is the formula used for this question? "Jamal wants to save $54,000 for a down payment on a home. How much will he need to invest in an account with 8.2% APR, compounding daily, in order to reach his goal in 5 years?"
Kuz Reply
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Kuz
A = P(1 + r/n) ^rt
Dale
how to solve an expression when equal to zero
Mintah Reply
its a very simple
Kavita
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Masum
What is the value of log-1
Masum
the value of log1=0
Kavita
Log(-1)
Masum
What is the value of i^i
Masum
log -1 is 1.36
kurash
No
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Kavita
tan20°×tan30°×tan45°×tan50°×tan60°×tan70°
Joju Reply
jaldi batao
Joju
Find the value of x between 0degree and 360 degree which satisfy the equation 3sinx =tanx
musah Reply
what is sine?
tae Reply
what is the standard form of 1
Sanjana Reply
1×10^0
Akugry
Evalute exponential functions
Sujata Reply
30
Shani
The sides of a triangle are three consecutive natural number numbers and it's largest angle is twice the smallest one. determine the sides of a triangle
Jaya Reply
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Inkoom
3, 4, 5 principle from geo? sounds like a 90 and 2 45's to me that my answer
Neese
answer is 2, 3, 4
Gaurav
prove that [a+b, b+c, c+a]= 2[a b c]
Ashutosh Reply
can't prove
Akugry
i can prove [a+b+b+c+c+a]=2[a+b+c]
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Stormzy
x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
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Oliver
proof for set theory
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find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
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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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