{"id":44,"date":"2017-08-23T07:10:07","date_gmt":"2017-08-23T07:10:07","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=44"},"modified":"2017-09-21T19:39:47","modified_gmt":"2017-09-21T19:39:47","slug":"relations-and-functions","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/relations-and-functions\/","title":{"rendered":"Relations and Functions"},"content":{"rendered":"
\n
\u2b05 Previous Lesson<\/a>\u00a0Workshop Index<\/a>\u00a0Next Lesson \u27a1<\/a><\/div>\n

<\/p>\n

Relations and Functions<\/h1>\n

Objective<\/h4>\n

In this lesson, you will study the definition and uses of relations and functions, including how to use tables,\u00a0graphs, verbal rules, and symbolic rules.<\/p>\n

\n

What is a relation?<\/strong><\/h3>\n

In mathematics, a relation<\/abbr> may be:<\/p>\n

    \n
  1. a rule or mapping between elements of two sets,<\/li>\n
  2. a set of ordered pairs, or<\/li>\n
  3. an equation or inequality in two variables.<\/li>\n<\/ol>\n

    The domain<\/abbr> of\u00a0a relation is the set of all possible values for the first element\u00a0in the mapping, first coordinate of the ordered pair, or\u00a0x-<\/em>variable\u00a0in an equation or inequality. The\u00a0range<\/abbr> of a relation is all the possible values of the second element,\u00a0coordinate, or y<\/em>-variable.<\/p>\n

    What is the\u00a0difference between a relation and a function?<\/p>\n

    A function<\/abbr> is a relation in which each element in the domain corresponds to\u00a0one, and only one, element in the range. Graphically, a curve\u00a0expresses a function if any vertical line passing through the\u00a0curve intersects the curve once only.<\/p>\n

    Relations and functions can be represented by tables, graphs,\u00a0rules, or equations.<\/p>\n

    For example, the following table represents a relation because\u00a0it is a set of ordered pairs. The domain is {1, 2, 3, 4}. The\u00a0range is {2, 3, 4, 5}. The table also represents a function\u00a0because each x-value has one and only one y-value.<\/p>\n\n\n\n\n\n\n\n\n
    x<\/th>\ny<\/th>\n<\/tr>\n<\/thead>\n
    1<\/td>\n2<\/td>\n<\/tr>\n
    2<\/td>\n3<\/td>\n<\/tr>\n
    3<\/td>\n4<\/td>\n<\/tr>\n
    4<\/td>\n5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
    \n

    Question<\/h4>\n
    \n

    Consider the set of ordered pairs {(6, 7) (8, 9) (10, 11) (12,\u00a013)}. This set of ordered pairs is best described as being<\/p>\n

      \n
    1. a relation only.<\/li>\n
    2. a function only.<\/li>\n
    3. a relation and a function.<\/li>\n
    4. neither a relation nor a function.<\/li>\n<\/ol>\n<\/div>\n

      Reveal Answer <\/a><\/p>\n

      \n

      The correct answer is C. The set is a relation because it is a\u00a0set of ordered pairs. The domain is {6, 8, 10, 12}. The range is\u00a0{7, 9, 11, 13}. The set is a function because each element in the\u00a0domain corresponds to one element in the range. These ordered\u00a0pairs can be derived from the equation\u00a0. Thus, we know that that equation is a function.<\/p>\n<\/div>\n<\/section>\n

      A function can be graphed in the following way.<\/p>\n

      <\/center>Note that any vertical line will intersect the function only\u00a0once.<\/p>\n
      \n

      Question<\/h4>\n
      \n

      Consider the graph below.<\/p>\n

      <\/center><\/a>This graph is best described as depicting<\/p>\n
        \n
      1. a relation only.<\/li>\n
      2. a function only.<\/li>\n
      3. a relation and a function.<\/li>\n
      4. neither a relation nor a function.<\/li>\n<\/ol>\n<\/div>\n

        Reveal Answer <\/a><\/p>\n

        \n

        The correct answer is C. First, we know it is a relation\u00a0because each point on the curve is an ordered pair. It is a\u00a0function because we cannot draw a vertical line that intersects\u00a0the curve more than once.<\/p>\n<\/div>\n<\/section>\n

        \n

        Question<\/h4>\n
        \n

        Consider the graph below.<\/p>\n

        <\/center><\/a>This graph is best described as depicting<\/p>\n
          \n
        1. a relation only.<\/li>\n
        2. a function only.<\/li>\n
        3. a relation and a function.<\/li>\n
        4. neither a relation nor a function.<\/li>\n<\/ol>\n<\/div>\n

          Reveal Answer <\/a><\/p>\n

          \n

          The correct answer is A. The graph depicts a relation, because\u00a0the circle is made up of points representing ordered pairs. The\u00a0curve is not a function, however, because you can draw vertical\u00a0lines through it that intersect it more than once.<\/p>\n<\/div>\n<\/section>\n

          What is functional notation?<\/h3>\n

          When writing a rule or an equation that represents a function,\u00a0we use functional\u00a0notation<\/abbr>. That means that we replace y<\/em> in the equation with \u0192(x).\u00a0<\/em>We can also use g(x)\u00a0<\/em>or h(x)<\/em>,\u00a0or any letter through m.
          \n<\/em><\/p>\n

          So,\u00a0\u00a0becomes \u00a0<\/em>in functional notation.<\/p>\n

          To show that ordered pairs are a function, use \u0192, g, or h\u00a0<\/span><\/em>alone.<\/p>\n

          <\/center><\/p>\n
          \n

          Question<\/h4>\n
          \n

          If ,\u00a0<\/em>which choice\u00a0ordered pair can be found for \u00a0?
          \n<\/em><\/p>\n

            \n
          1. (\u20132, 5)<\/li>\n
          2. (2, 5)<\/li>\n
          3. ( \u20132, \u20133)<\/li>\n
          4. (5, \u20132)<\/li>\n<\/ol>\n<\/div>\n

            Reveal Answer <\/a><\/p>\n

            \n

            The correct answer is A. To find the ordered pair, substitute\u00a0\u20132 for x\u00a0<\/em>and solve for f<\/em>(x<\/em>).<\/span><\/span><\/p>\n

            <\/p>\n<\/div>\n<\/section>\n

            Is the relation in the example above a function?<\/h4>\n

            To find out, check whether any value of\u00a0x\u00a0<\/em>produces more than one value for \u0192(x<\/em>). Find out by testing various values, both positive and negative,\u00a0for x<\/em>. In this case, no x value can produce more than one value for\u00a0\u0192(x<\/em>),\u00a0so the relation is a function. Note, however, that there will be\u00a0two values for x<\/em>, a positive one and a negative one, that map to any value\u00a0of \u0192(x).<\/p>\n

            What are the types of functions?<\/h4>\n

            A linear\u00a0function<\/abbr> is a function that can be written in the\u00a0form\u00a0,\u00a0<\/em>where m <\/em>and b <\/em>are real numbers and m \u2260 0. <\/em>For\u00a0a function in this form, y, or \u0192(x)<\/em> is a dependent\u00a0variable<\/abbr>, and x<\/em> is an independent\u00a0variable<\/abbr>.<\/p>\n

            The simplest form of a function is called a parent\u00a0function<\/abbr>. For linear functions,\u00a0\u00a0is the parent function.<\/p>\n

            \n

            Important Tidbit<\/h4>\n

            is the parent function for quadratic functions<\/abbr>.<\/p>\n

            is the parent function of exponential functions<\/abbr>.<\/p>\n<\/div>\n<\/section>\n

            <\/p>\n

            \u2b05 Previous Lesson<\/a>\u00a0Workshop Index<\/a>\u00a0Next Lesson \u27a1<\/a><\/div>\n

            Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

            \u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Lesson \u27a1 Relations and Functions Objective In this lesson, you will study the definition and uses of relations and functions, including how to use tables,\u00a0graphs, verbal rules, and symbolic rules. What is a relation? In mathematics, a relation may be: a rule or mapping between elements of two sets, a set […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-44","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/44","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=44"}],"version-history":[{"count":10,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/44\/revisions"}],"predecessor-version":[{"id":787,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/44\/revisions\/787"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=44"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}