{"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":"2018-06-18T18:54:03","modified_gmt":"2018-06-18T18:54:03","slug":"determine-whether-a-function-is-one-to-one","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/chapter\/determine-whether-a-function-is-one-to-one\/","title":{"raw":"Determine whether a function is one-to-one","rendered":"Determine whether a function is one-to-one"},"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 13: 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<\/div>\r\n
\r\n

Solution<\/h3>\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

If 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.<\/p>\r\n\r\n<\/div>\r\n

\r\n

Try It 7<\/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\nSolution<\/a>\r\n\r\n<\/div>\r\n<\/section>","rendered":"
    \n
    \"Figure<\/p>\n

    Figure 9<\/b><\/p>\n<\/div>\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>\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>\n\n\n\n<\/colgroup>\n\n\n\n\n\n\n\n
    Letter grade<\/th>\nGrade point average<\/th>\n<\/tr>\n<\/thead>\n
    A<\/td>\n4.0<\/td>\n<\/tr>\n
    B<\/td>\n3.0<\/td>\n<\/tr>\n
    C<\/td>\n2.0<\/td>\n<\/tr>\n
    D<\/td>\n1.0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\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>\n

    To visualize this concept, let\u2019s look again at the two simple functions sketched in (a)and (b) of Figure 10.<\/p>\n

    \"Three<\/p>\n

    Figure 10<\/b><\/p>\n<\/div>\n

    The 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.<\/p>\n

    \n

    A General Note: One-to-One Function<\/h3>\n

    A one-to-one function is a function in which each output value corresponds to exactly one input value.<\/p>\n<\/div>\n

    \n

    Example 13: Determining Whether a Relationship Is a One-to-One Function<\/h3>\n

    Is the area of a circle a function of its radius? If yes, is the function one-to-one?<\/p>\n<\/div>\n

    \n

    Solution<\/h3>\n

    A 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].<\/p>\n

    If 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.<\/p>\n<\/div>\n

    \n

    Try It 7<\/h3>\n
      \n
    1. Is a balance a function of the bank account number?<\/li>\n
    2. Is a bank account number a function of the balance?<\/li>\n
    3. Is a balance a one-to-one function of the bank account number?<\/li>\n<\/ol>\n

      Solution<\/a><\/p>\n<\/div>\n<\/section>\n\n\t\t\t

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