{"id":1904,"date":"2024-07-25T01:04:55","date_gmt":"2024-07-25T01:04:55","guid":{"rendered":"https:\/\/www.methodologydesign.com\/?p=1904"},"modified":"2025-02-14T20:59:23","modified_gmt":"2025-02-14T20:59:23","slug":"lesson-1-1-1-the-identification-and-assessment-of-functions","status":"publish","type":"post","link":"https:\/\/www.methodologydesign.com\/index.php\/2024\/07\/25\/lesson-1-1-1-the-identification-and-assessment-of-functions\/","title":{"rendered":"Lesson 1.1.1 – The Identification and Assessment of Functions"},"content":{"rendered":"Reading Time: <\/span> 3<\/span> minutes<\/span><\/span>[et_pb_section admin_label=”section”]\n\t\t\t[et_pb_row admin_label=”row”]\n\t\t\t\t[et_pb_column type=”4_4″][et_pb_text admin_label=”Text”]\n

Overview of Covered Topics<\/h3>\n\n\n\n

In this lesson, the relationships between two variables, specifically functions, will be revisited. The lesson will address the following key points:<\/p>\n\n\n\n

    \n
  1. Determining Whether a Relation Represents a Function<\/li>\n\n\n\n
  2. Functions Defined by Sets of Ordered Pairs<\/li>\n\n\n\n
  3. Functions Defined by Equations<\/li>\n\n\n\n
  4. Finding Input and Output Values of a Function<\/li>\n\n\n\n
  5. The Importance of Function Notation<\/li>\n<\/ol>\n\n\n\n

    Determining Whether a Relation Represents a Function<\/h4>\n\n\n\n

    In mathematics, a function is a specific type of relationship where each input is associated with only one output. This concept is essential, as the output is determined by the input(s).<\/p>\n\n\n\n

    Examples of familiar functions include:<\/p>\n\n\n\n

      \n
    • Area of a Rectangle<\/strong>: A=LWA = LWA=LW (Inputs: L=L =L= Length, W=W =W= Width; Output: A=A =A= Area)<\/li>\n\n\n\n
    • Perimeter of a Square<\/strong>: P=4SP = 4SP=4S (Input: S=S =S= Side Length; Output: P=P =P= Perimeter)<\/li>\n\n\n\n
    • Volume of a Right Circular Cylinder<\/strong>: V=\u03c0r2hV = \\pi r^2 hV=\u03c0r2h (Inputs: r=r =r= Radius, h=h =h= Height; Output: V=V =V= Volume)<\/li>\n<\/ul>\n\n\n\n

      To further understand the concept of a function, consider a calculator: when the square root key is pressed, it returns the square root of the input (e.g., input 16, output 4). The calculator’s consistent output for each input exemplifies a function.<\/p>\n\n\n\n

      Term to Know<\/strong>: Function<\/em> – A correspondence between a set of inputs, xxx, and a set of outputs, yyy, where each input corresponds to at most one output.<\/p>\n\n\n\n

      Functions Defined by Sets of Ordered Pairs<\/h5>\n\n\n\n

      A function can be represented by a collection of ordered pairs (x,y)(x, y)(x,y), where xxx is the input and yyy is the output.<\/p>\n\n\n\n

      Example<\/strong>:<\/p>\n\n\n\n

        \n
      • Let xxx represent the temperature in Celsius and yyy the corresponding temperature in Fahrenheit. Examples of ordered pairs: (0,32)(0, 32)(0,32), (100,212)(100, 212)(100,212), (37,98.6)(37, 98.6)(37,98.6), (\u221240,\u221240)(-40, -40)(\u221240,\u221240). This relationship is a function since each Celsius temperature has a unique Fahrenheit equivalent.<\/li>\n<\/ul>\n\n\n\n

        Another example involving students and homework:<\/p>\n\n\n\n

          \n
        • Let xxx be the number of assignments completed and yyy the quiz score. Ordered pairs for several students might be {(4,15),(3,12),(5,18),(4,10),(3,16),(2,10),(5,14),(4,16)}\\{(4, 15), (3, 12), (5, 18), (4, 10), (3, 16), (2, 10), (5, 14), (4, 16)\\}{(4,15),(3,12),(5,18),(4,10),(3,16),(2,10),(5,14),(4,16)}. This relationship is not a function since some inputs (e.g., x=4x = 4x=4) correspond to multiple outputs (15, 10, 16).<\/li>\n<\/ul>\n\n\n\n
          Functions Defined by Equations<\/h5>\n\n\n\n

          This course primarily deals with equations involving two variables. Generally, xxx is the input variable and yyy the output variable. When an equation defines a function, yyy is a function of xxx.<\/p>\n\n\n\n

          Examples<\/strong>:<\/p>\n\n\n\n

            \n
          • y=2x+3y = 2x + 3y=2x+3: This is a function as each input xxx yields a unique output yyy. Example ordered pairs: (0,3)(0, 3)(0,3), (1,5)(1, 5)(1,5), (2,7)(2, 7)(2,7), (3,9)(3, 9)(3,9).<\/li>\n\n\n\n
          • x2+y2=25x^2 + y^2 = 25×2+y2=25: This is not a function because some xxx-values correspond to multiple yyy-values, e.g., (0,5)(0, 5)(0,5) and (0,\u22125)(0, -5)(0,\u22125).<\/li>\n\n\n\n
          • y=x2\u22124x\u22122y = x^2 – 4x – 2y=x2\u22124x\u22122: This is a function as each xxx corresponds to a single yyy. Example ordered pairs: (1,\u22125)(1, -5)(1,\u22125), (2,\u22126)(2, -6)(2,\u22126), (3,\u22125)(3, -5)(3,\u22125), (4,\u22122)(4, -2)(4,\u22122).<\/li>\n<\/ul>\n\n\n\n

            Big Idea<\/strong>: Any equation that can be written without using the plus-or-minus sign (\u00b1\\pm\u00b1) defines a function.<\/p>\n\n\n\n

            Example<\/strong>: For the equation 2x\u22123y=62x – 3y = 62x\u22123y=6, solving for yyy gives y=23x\u22122y = \\frac{2}{3}x – 2y=32\u200bx\u22122. Since there is no plus-or-minus, yyy is a function of xxx.<\/p>\n\n\n\n

            Finding Input and Output Values of a Function<\/h4>\n\n\n\n

            When a relation qualifies as a function, it can be expressed using function notation. For instance, y=x2y = x^2y=x2 can be written as f(x)=x2f(x) = x^2f(x)=x2. Here, xxx is the input variable, and f(x)f(x)f(x) is the output variable.<\/p>\n\n\n\n

            Example<\/strong>: For f(x)=x2f(x) = x^2f(x)=x2, find f(2)f(2)f(2), f(\u22124)f(-4)f(\u22124), and f(a+3)f(a+3)f(a+3):<\/p>\n\n\n\n

              \n
            • f(2)=22=4f(2) = 2^2 = 4f(2)=22=4<\/li>\n\n\n\n
            • f(\u22124)=(\u22124)2=16f(-4) = (-4)^2 = 16f(\u22124)=(\u22124)2=16<\/li>\n\n\n\n
            • f(a+3)=(a+3)2=a2+6a+9f(a+3) = (a+3)^2 = a^2 + 6a + 9f(a+3)=(a+3)2=a2+6a+9<\/li>\n<\/ul>\n\n\n\n

              Hint<\/strong>: Ensure the entire input is squared.<\/p>\n\n\n\n

              Try It<\/strong>: Use f(x)=x2\u22124x+2f(x) = x^2 – 4x + 2f(x)=x2\u22124x+2 to find f(1)f(1)f(1), f(\u22123)f(-3)f(\u22123), and f(a+1)f(a+1)f(a+1).<\/p>\n\n\n\n

              The Importance of Function Notation<\/h4>\n\n\n\n

              Function notation is useful for computing different values from a single input.<\/p>\n\n\n\n

              Example<\/strong>: For a square with side length xxx:<\/p>\n\n\n\n

                \n
              • Area: A(x)=x2A(x) = x^2A(x)=x2<\/li>\n\n\n\n
              • Perimeter: P(x)=4xP(x) = 4xP(x)=4x<\/li>\n\n\n\n
              • Diagonal: D(x)=x2D(x) = x\\sqrt{2}D(x)=x2\u200b<\/li>\n<\/ul>\n\n\n\n

                For a square of side 5 inches:<\/p>\n\n\n\n

                  \n
                • Area: A(5)=25A(5) = 25A(5)=25 square inches<\/li>\n\n\n\n
                • Perimeter: P(5)=20P(5) = 20P(5)=20 inches<\/li>\n\n\n\n
                • Diagonal: D(5)=52D(5) = 5\\sqrt{2}D(5)=52\u200b inches<\/li>\n<\/ul>\n\n\n\n

                  Function names (A, P, D) provide meaningful output names beyond yyy.<\/p>\n\n\n\n

                  Hint<\/strong>: The input variable can vary, e.g., height of a projectile over time h(t)h(t)h(t).<\/p>\n\n\n\n

                  Summary<\/h3>\n\n\n\n

                  This lesson covered the determination of whether a relation is a function, emphasizing that each input must correspond to only one output. Functions can be defined by sets of ordered pairs (x,y)(x, y)(x,y) or equations where yyy is a function of xxx. Function notation highlights the relationship between input and output values, with xxx as the input variable and f(x)f(x)f(x) as the output variable.<\/p>\n[\/et_pb_text][\/et_pb_column]\n\t\t\t[\/et_pb_row]\n\t\t[\/et_pb_section]","protected":false},"excerpt":{"rendered":"

                  Reading Time: <\/span> 3<\/span> minutes<\/span><\/span><\/p>\n

                  Overview of Covered Topics In this lesson, the relationships between two variables, specifically functions, will be revisited. The lesson will address the following key points: Determining Whether a Relation Represents a Function Functions Defined by Sets of Ordered Pairs Functions Defined by Equations Finding Input and Output Values of a Function The Importance of Function […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_et_pb_use_builder":"on","_et_pb_old_content":"\n

                  Overview of Covered Topics<\/h3>\n\n\n\n

                  In this lesson, the relationships between two variables, specifically functions, will be revisited. The lesson will address the following key points:<\/p>\n\n\n\n

                    \n
                  1. Determining Whether a Relation Represents a Function<\/li>\n\n\n\n
                  2. Functions Defined by Sets of Ordered Pairs<\/li>\n\n\n\n
                  3. Functions Defined by Equations<\/li>\n\n\n\n
                  4. Finding Input and Output Values of a Function<\/li>\n\n\n\n
                  5. The Importance of Function Notation<\/li>\n<\/ol>\n\n\n\n

                    Determining Whether a Relation Represents a Function<\/h4>\n\n\n\n

                    In mathematics, a function is a specific type of relationship where each input is associated with only one output. This concept is essential, as the output is determined by the input(s).<\/p>\n\n\n\n

                    Examples of familiar functions include:<\/p>\n\n\n\n

                      \n
                    • Area of a Rectangle<\/strong>: A=LWA = LWA=LW (Inputs: L=L =L= Length, W=W =W= Width; Output: A=A =A= Area)<\/li>\n\n\n\n
                    • Perimeter of a Square<\/strong>: P=4SP = 4SP=4S (Input: S=S =S= Side Length; Output: P=P =P= Perimeter)<\/li>\n\n\n\n
                    • Volume of a Right Circular Cylinder<\/strong>: V=\u03c0r2hV = pi r^2 hV=\u03c0r2h (Inputs: r=r =r= Radius, h=h =h= Height; Output: V=V =V= Volume)<\/li>\n<\/ul>\n\n\n\n

                      To further understand the concept of a function, consider a calculator: when the square root key is pressed, it returns the square root of the input (e.g., input 16, output 4). The calculator's consistent output for each input exemplifies a function.<\/p>\n\n\n\n

                      Term to Know<\/strong>: Function<\/em> - A correspondence between a set of inputs, xxx, and a set of outputs, yyy, where each input corresponds to at most one output.<\/p>\n\n\n\n

                      Functions Defined by Sets of Ordered Pairs<\/h5>\n\n\n\n

                      A function can be represented by a collection of ordered pairs (x,y)(x, y)(x,y), where xxx is the input and yyy is the output.<\/p>\n\n\n\n

                      Example<\/strong>:<\/p>\n\n\n\n

                        \n
                      • Let xxx represent the temperature in Celsius and yyy the corresponding temperature in Fahrenheit. Examples of ordered pairs: (0,32)(0, 32)(0,32), (100,212)(100, 212)(100,212), (37,98.6)(37, 98.6)(37,98.6), (\u221240,\u221240)(-40, -40)(\u221240,\u221240). This relationship is a function since each Celsius temperature has a unique Fahrenheit equivalent.<\/li>\n<\/ul>\n\n\n\n

                        Another example involving students and homework:<\/p>\n\n\n\n

                          \n
                        • Let xxx be the number of assignments completed and yyy the quiz score. Ordered pairs for several students might be {(4,15),(3,12),(5,18),(4,10),(3,16),(2,10),(5,14),(4,16)}{(4, 15), (3, 12), (5, 18), (4, 10), (3, 16), (2, 10), (5, 14), (4, 16)}{(4,15),(3,12),(5,18),(4,10),(3,16),(2,10),(5,14),(4,16)}. This relationship is not a function since some inputs (e.g., x=4x = 4x=4) correspond to multiple outputs (15, 10, 16).<\/li>\n<\/ul>\n\n\n\n
                          Functions Defined by Equations<\/h5>\n\n\n\n

                          This course primarily deals with equations involving two variables. Generally, xxx is the input variable and yyy the output variable. When an equation defines a function, yyy is a function of xxx.<\/p>\n\n\n\n

                          Examples<\/strong>:<\/p>\n\n\n\n

                            \n
                          • y=2x+3y = 2x + 3y=2x+3: This is a function as each input xxx yields a unique output yyy. Example ordered pairs: (0,3)(0, 3)(0,3), (1,5)(1, 5)(1,5), (2,7)(2, 7)(2,7), (3,9)(3, 9)(3,9).<\/li>\n\n\n\n
                          • x2+y2=25x^2 + y^2 = 25x2+y2=25: This is not a function because some xxx-values correspond to multiple yyy-values, e.g., (0,5)(0, 5)(0,5) and (0,\u22125)(0, -5)(0,\u22125).<\/li>\n\n\n\n
                          • y=x2\u22124x\u22122y = x^2 - 4x - 2y=x2\u22124x\u22122: This is a function as each xxx corresponds to a single yyy. Example ordered pairs: (1,\u22125)(1, -5)(1,\u22125), (2,\u22126)(2, -6)(2,\u22126), (3,\u22125)(3, -5)(3,\u22125), (4,\u22122)(4, -2)(4,\u22122).<\/li>\n<\/ul>\n\n\n\n

                            Big Idea<\/strong>: Any equation that can be written without using the plus-or-minus sign (\u00b1pm\u00b1) defines a function.<\/p>\n\n\n\n

                            Example<\/strong>: For the equation 2x\u22123y=62x - 3y = 62x\u22123y=6, solving for yyy gives y=23x\u22122y = frac{2}{3}x - 2y=32\u200bx\u22122. Since there is no plus-or-minus, yyy is a function of xxx.<\/p>\n\n\n\n

                            Finding Input and Output Values of a Function<\/h4>\n\n\n\n

                            When a relation qualifies as a function, it can be expressed using function notation. For instance, y=x2y = x^2y=x2 can be written as f(x)=x2f(x) = x^2f(x)=x2. Here, xxx is the input variable, and f(x)f(x)f(x) is the output variable.<\/p>\n\n\n\n

                            Example<\/strong>: For f(x)=x2f(x) = x^2f(x)=x2, find f(2)f(2)f(2), f(\u22124)f(-4)f(\u22124), and f(a+3)f(a+3)f(a+3):<\/p>\n\n\n\n

                              \n
                            • f(2)=22=4f(2) = 2^2 = 4f(2)=22=4<\/li>\n\n\n\n
                            • f(\u22124)=(\u22124)2=16f(-4) = (-4)^2 = 16f(\u22124)=(\u22124)2=16<\/li>\n\n\n\n
                            • f(a+3)=(a+3)2=a2+6a+9f(a+3) = (a+3)^2 = a^2 + 6a + 9f(a+3)=(a+3)2=a2+6a+9<\/li>\n<\/ul>\n\n\n\n

                              Hint<\/strong>: Ensure the entire input is squared.<\/p>\n\n\n\n

                              Try It<\/strong>: Use f(x)=x2\u22124x+2f(x) = x^2 - 4x + 2f(x)=x2\u22124x+2 to find f(1)f(1)f(1), f(\u22123)f(-3)f(\u22123), and f(a+1)f(a+1)f(a+1).<\/p>\n\n\n\n

                              The Importance of Function Notation<\/h4>\n\n\n\n

                              Function notation is useful for computing different values from a single input.<\/p>\n\n\n\n

                              Example<\/strong>: For a square with side length xxx:<\/p>\n\n\n\n

                                \n
                              • Area: A(x)=x2A(x) = x^2A(x)=x2<\/li>\n\n\n\n
                              • Perimeter: P(x)=4xP(x) = 4xP(x)=4x<\/li>\n\n\n\n
                              • Diagonal: D(x)=x2D(x) = xsqrt{2}D(x)=x2\u200b<\/li>\n<\/ul>\n\n\n\n

                                For a square of side 5 inches:<\/p>\n\n\n\n

                                  \n
                                • Area: A(5)=25A(5) = 25A(5)=25 square inches<\/li>\n\n\n\n
                                • Perimeter: P(5)=20P(5) = 20P(5)=20 inches<\/li>\n\n\n\n
                                • Diagonal: D(5)=52D(5) = 5sqrt{2}D(5)=52\u200b inches<\/li>\n<\/ul>\n\n\n\n

                                  Function names (A, P, D) provide meaningful output names beyond yyy.<\/p>\n\n\n\n

                                  Hint<\/strong>: The input variable can vary, e.g., height of a projectile over time h(t)h(t)h(t).<\/p>\n\n\n\n

                                  Summary<\/h3>\n\n\n\n

                                  This lesson covered the determination of whether a relation is a function, emphasizing that each input must correspond to only one output. Functions can be defined by sets of ordered pairs (x,y)(x, y)(x,y) or equations where yyy is a function of xxx. Function notation highlights the relationship between input and output values, with xxx as the input variable and f(x)f(x)f(x) as the output variable.<\/p>\n","_et_gb_content_width":"","footnotes":""},"categories":[118],"tags":[],"class_list":["post-1904","post","type-post","status-publish","format-standard","hentry","category-precalculus-1"],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/www.methodologydesign.com\/index.php\/wp-json\/wp\/v2\/posts\/1904","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.methodologydesign.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.methodologydesign.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.methodologydesign.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.methodologydesign.com\/index.php\/wp-json\/wp\/v2\/comments?post=1904"}],"version-history":[{"count":2,"href":"https:\/\/www.methodologydesign.com\/index.php\/wp-json\/wp\/v2\/posts\/1904\/revisions"}],"predecessor-version":[{"id":1945,"href":"https:\/\/www.methodologydesign.com\/index.php\/wp-json\/wp\/v2\/posts\/1904\/revisions\/1945"}],"wp:attachment":[{"href":"https:\/\/www.methodologydesign.com\/index.php\/wp-json\/wp\/v2\/media?parent=1904"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.methodologydesign.com\/index.php\/wp-json\/wp\/v2\/categories?post=1904"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.methodologydesign.com\/index.php\/wp-json\/wp\/v2\/tags?post=1904"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}