{"id":10710,"date":"2015-07-10T19:05:10","date_gmt":"2015-07-10T19:05:10","guid":{"rendered":"https:\/\/courses.candelalearning.com\/osprecalc\/?post_type=chapter&p=10710"},"modified":"2021-10-25T01:57:58","modified_gmt":"2021-10-25T01:57:58","slug":"one-to-one-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/one-to-one-functions\/","title":{"raw":"One-to-one functions","rendered":"One-to-one functions"},"content":{"raw":"
\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]\"Figure Figure 9<\/b>[\/caption]\r\n

Some functions have a given output value that corresponds to two or more input values. For example, in the following stock chart the stock price was $1000 on five different dates, meaning that there were five different input values that all resulted in the same output value of $1000.<\/p>\r\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.<\/p>\r\n\r\n<\/colgroup>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Letter grade<\/th>\r\nGrade point average<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
A<\/td>\r\n4.0<\/td>\r\n<\/tr>\r\n
B<\/td>\r\n3.0<\/td>\r\n<\/tr>\r\n
C<\/td>\r\n2.0<\/td>\r\n<\/tr>\r\n
D<\/td>\r\n1.0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

This grading system represents a one-to-one function, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.<\/p>\r\nTo visualize this concept, let\u2019s look again at the two simple functions sketched in (a)and (b) of Figure 10.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]\"Three Figure 10<\/b>[\/caption]\r\n\r\nThe function in part (a) shows a relationship that is not a one-to-one function because inputs [latex]q[\/latex] and [latex]r[\/latex] both give output [latex]n[\/latex]. The function in part (b) shows a relationship that is a one-to-one function because each input is associated with a single output.\r\n

\r\n

A General Note: One-to-One Function<\/h3>\r\nA one-to-one function is a function in which each output value corresponds to exactly one input value.\r\n\r\n<\/div>\r\n
\r\n

Example: Determining Whether a Relationship Is a One-to-One Function<\/h3>\r\nIs the area of a circle a function of its radius? If yes, is the function one-to-one?\r\n\r\n[reveal-answer q=\"380432\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"380432\"]\r\n\r\nA circle of radius [latex]r[\/latex] has a unique area measure given by [latex]A=\\pi {r}^{2}[\/latex], so for any input, [latex]r[\/latex], there is only one output, [latex]A[\/latex]. The area is a function of radius [latex]r[\/latex].\r\n\r\nIf the function is one-to-one, the output value, the area, must correspond to a unique input value, the radius. Any area measure [latex]A[\/latex] is given by the formula [latex]A=\\pi {r}^{2}[\/latex]. Because areas and radii are positive numbers, there is exactly one solution: [latex]r=\\sqrt{\\frac{A}{\\pi }}[\/latex]. So the area of a circle is a one-to-one function of the circle\u2019s radius.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n
\r\n

Try It<\/h3>\r\n
    \r\n \t
  1. Is a balance a function of the bank account number?<\/li>\r\n \t
  2. Is a bank account number a function of the balance?<\/li>\r\n \t
  3. Is a balance a one-to-one function of the bank account number?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"997233\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"997233\"]\r\n
      \r\n \t
    1. yes, because each bank account (input) has a single balance (output) at any given time.<\/span><\/li>\r\n \t
    2. no, because several bank accounts (inputs) may have the same balance (output).<\/span><\/li>\r\n \t
    3. no, because the more than one bank account (input) can have the same balance (output).<\/span><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section>\r\n

      The Horizontal Line Test<\/h2>\r\nOnce 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. A horizontal line includes all points with a particular [latex]y[\/latex] value. The [latex]x[\/latex] value of a point where a vertical line intersects a function represents the input for that output [latex]y[\/latex] value. 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.\r\n
      \r\n

      How To: Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function.<\/h3>\r\n
        \r\n \t
      1. Inspect the graph to see if any horizontal line drawn would intersect the curve more than once.<\/li>\r\n \t
      2. If there is any such line, the function is not one-to-one.<\/li>\r\n \t
      3. If no horizontal line can intersect the curve more than once, the function is one-to-one.<\/li>\r\n<\/ol>\r\n<\/div>\r\n
        \r\n

        Example: Applying the Horizontal Line Test<\/h3>\r\nConsider the functions (a), and (b)shown in\u00a0the graphs\u00a0below.\r\n\r\n\"Graph\r\n\r\nAre either of the functions one-to-one?\r\n\r\n[reveal-answer q=\"173050\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"173050\"]\r\n\r\nThe 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.)\r\n\r\n\r\n\r\nThe function in (b) is one-to-one. Any horizontal line will intersect a diagonal line at most once.\r\n\r\n[\/hidden-answer]\r\n