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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

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Astronomy (from Ancient Greek ἀστρονομία (astronomía) 'science that studies the laws of the stars') is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution.
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the big bang theory is a theory which states that all matter was compressed together in one place the matter got so unstable it exploded releasing All its contents in the form of hydrogen
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solar Univers
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bigbang is the beginning of the universe
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but thats just a theory
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these are Rocky substances between mars and jupiter
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Comets are cosmic snowballs of frozen gases , rock and dust that orbit the sun. They are mostly found between the orbits of Venus and Mercury.
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how can we learn right and true ?
Govinda Reply
why the moon is always appear in an elliptical shape
Gatjuol Reply
Because when astroid hit the Earth then a piece of elliptical shape of the earth was separated which is now called moon.
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what's see level?
lidiya Reply
Did you mean eye sight or sea level
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according to the theory of astronomers why the moon is always appear in an elliptical orbit?
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we're all stars and galaxies a part of sun. how can science prove thx with respect old ancient times picture or books..or anything with respect to present time .but we r a part of that universe
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there many theory to born universe but what is the reality of big bang theory to born universe
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there are many theories regarding this it's on you believe any theory that you think is true ex. eternal inflation theory, oscillation model theory, multiple universe theory the big bang theory etc.
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I think it couldn't possible on earth
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is that so. the question was in the end of this chapter
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in theory, you could see them all from the equator (though over the course of a year, not at pne time). stars are measured in "declination", which is how far N or S of the equator (90* to -90*). Polaris is the North star, and is ALMOST 90* (+89*). So it would just barely creep over the horizon.
Christopher
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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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