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For example, the black dots on the graph above tell us that [latex]f(0)=2[\/latex] and [latex]f(6)=1[\/latex]. However, the set of all points [latex](x,y)[\/latex] satisfying [latex]y=f(x)[\/latex] is a curve. The curve shown includes [latex](0,2)[\/latex] and [latex](6,1)[\/latex] because the curve passes through those points.<\/p>\n
The vertical line test<\/strong> 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.<\/p>\n The vertical line test<\/strong> determines if a relation is a function by checking that no vertical line intersects the graph more than once.<\/p>\n<\/section>\n How to: Use the Vertical Line Test<\/strong> [ohm2_question hide_question_numbers=1]13503[\/ohm2_question]<\/p>\n<\/section>\n Some functions have a given output value that corresponds to two or more input values. For example, in the stock chart shown below, the stock price was [latex]$1000[\/latex] on five different dates, meaning that there were five different input values that all resulted in the same output value of [latex]$1000[\/latex].<\/p>\n <\/p>\n However, some functions have only one input value for each output value, as well as having only one output for each input. We call these functions one-to-one functions. As an example, consider a school that uses only letter grades and decimal equivalents, as listed in the table below.<\/p>\n This grading system represents a one-to-one function<\/strong>, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.<\/p>\n A one-to-one function<\/strong> is a function in which each output value corresponds to exactly one input value.<\/p>\n<\/div>\n<\/section>\n [ohm2_question hide_question_numbers=1]13502[\/ohm2_question]<\/p>\n<\/section>\n 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<\/strong>. Draw horizontal lines through the graph. If we can draw any<\/em> horizontal line that intersects a graph more than once, then the graph does not<\/em> represent a one-to-one function because that [latex]y[\/latex] value has more than one input.<\/p>\n The horizontal line test<\/strong> checks if a function is one-to-one by ensuring that no horizontal line intersects the graph more than once.<\/p>\n<\/section>\n How To: Horizontal Line Test<\/strong><\/p>\n Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function following these steps.<\/p>\n <\/p>\n\n\nThe function in (b) is one-to-one. Any horizontal line will intersect a diagonal line at most once. The function in (c) is not one-to-one. The horizontal line intersects the graph of the function at two points. [\/hidden-answer]<\/section>\n [ohm2_question hide_question_numbers=1]13504[\/ohm2_question]<\/p>\n<\/section>\n","rendered":" We can represent a function using a graph. Graphs display a great many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. By convention, graphs are typically constructed with the input values along the horizontal axis and the output values along the vertical axis.<\/p>\n The most common graphs name the input value [latex]x[\/latex] and the output value [latex]y[\/latex], and we say [latex]y[\/latex] is a function of [latex]x[\/latex], or [latex]y=f(x)[\/latex] when the function is named [latex]f[\/latex].<\/p>\n The graph of the function is the set of all points [latex](x,y)[\/latex] in the plane that satisfies the equation [latex]y=f(x)[\/latex]. If the function is defined for only a few input values, then the graph of the function is only a few points, where the [latex]x[\/latex]-coordinate of each point is an input value and the [latex]y[\/latex]-coordinate of each point is the corresponding output value.<\/p>\n <\/p>\n For example, the black dots on the graph above tell us that [latex]f(0)=2[\/latex] and [latex]f(6)=1[\/latex]. However, the set of all points [latex](x,y)[\/latex] satisfying [latex]y=f(x)[\/latex] is a curve. The curve shown includes [latex](0,2)[\/latex] and [latex](6,1)[\/latex] because the curve passes through those points.<\/p>\n The vertical line test<\/strong> 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.<\/p>\n The vertical line test<\/strong> determines if a relation is a function by checking that no vertical line intersects the graph more than once.<\/p>\n<\/section>\n How to: Use the Vertical Line Test<\/strong><\/p>\n Given a graph, use the vertical line test to determine if the graph represents a function following these steps.<\/p>\nvertical line test<\/h3>\n
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\nGiven a graph, use the vertical line test to determine if the graph represents a function following these steps.<\/p>\n\n\t
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[reveal-answer q=\"689864\"]Show Solution[\/reveal-answer] [hidden-answer a=\"689864\"] 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 the graph above. From this we can conclude that these two graphs represent functions. The third graph does not represent a function because, at most [latex]x[\/latex]-values, a vertical line would intersect the graph at more than one point.![\"Graph](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191020\/CNX_Precalc_Figure_01_01_016.jpg\")
<\/center>[\/hidden-answer]<\/section>\nDetermining Whether a Function is One-to-One<\/h2>\n
![\"a](\"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-content\/uploads\/sites\/18\/2023\/10\/be97793ae3e427723869ee41e52cd09a3286f4c1.png#fixme\")
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\n \n \n \nLetter grade<\/th>\n\n \n Grade point average<\/th>\n\n <\/tr>\n\n <\/thead>\n\n \n \n \n A<\/td>\n\n \n 4.0<\/td>\n\n <\/tr>\n\n \n \n \n B<\/td>\n\n \n 3.0<\/td>\n\n <\/tr>\n\n \n \n \n C<\/td>\n\n \n 2.0<\/td>\n\n <\/tr>\n\n \n \n \n D<\/td>\n\n \n 1.0<\/td>\n\n <\/tr>\n\n <\/tbody>\n<\/table>\n one-to-one function<\/h3>\n
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The Horizontal Line Test<\/h2>\n
horizontal line test<\/h3>\n
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[reveal-answer q=\"173050\"]Show Solution[\/reveal-answer] [hidden-answer a=\"173050\"] The function in (a) is not one-to-one. The horizontal line shown below intersects the graph of the function at two points (and we can even find horizontal lines that intersect it at three points.)![](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191025\/CNX_Precalc_Figure_01_01_010.jpg\")
<\/center>\nVerifying a Function Using the Vertical Line Test<\/h2>\n
<\/div>\nvertical line test<\/h3>\n
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