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Notation and Alg= ebra of Functions
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Notation and Alg= ebra of Functions
Definition: A <= span lang=3DEN-US style=3D'font-size:156%;color:red;mso-fareast-language:ZH-CN'= >function f is a procedure that assigns each input x from a se= t D (the domain), exactly one output y in a set R (t= he range). 
------=_NextPart_01C82E62.72341DF0 Content-Location: file:///C:/4D764A33/2NotationandAlgebraofFunctions_files/slide0139.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
Notation and Alg= ebra of Functions
Definition: A <= span lang=3DEN-US style=3D'font-size:156%;color:red;mso-fareast-language:ZH-CN'= >function f is a procedure that assigns each input x from a se= t D (the domain), exactly one output y in a set R (t= he range).  We write it as f(x) = =3D y,
where x = =3D input, y =3D output,
------=_NextPart_01C82E62.72341DF0 Content-Location: file:///C:/4D764A33/2NotationandAlgebraofFunctions_files/slide0140.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
Notation and Alg= ebra of Functions
Definition: A <= span lang=3DEN-US style=3D'font-size:156%;color:red;mso-fareast-language:ZH-CN'= >function f is a procedure that assigns each input x from a se= t D (the domain), exactly one output y in a set R (t= he range).  We write it as f(x) = =3D y,
where x = =3D input, y =3D output, or as f: D à R.
------=_NextPart_01C82E62.72341DF0 Content-Location: file:///C:/4D764A33/2NotationandAlgebraofFunctions_files/slide0141.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
Example A:
a. Input a number x, the output is (are)
whole number(s) within ¾ of x. Is this a function?
Notation and Alg= ebra of Functions
Definition: A <= span lang=3DEN-US style=3D'font-size:156%;color:red;mso-fareast-language:ZH-CN'= >function f is a procedure that assigns each input x from a se= t D (the domain), exactly one output y in a set R (t= he range).  We write it as f(x) = =3D y,
where x = =3D input, y =3D output, or as f: D à R.
------=_NextPart_01C82E62.72341DF0 Content-Location: file:///C:/4D764A33/2NotationandAlgebraofFunctions_files/slide0142.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
Example A:
a. Input a number x, the output is (are)
whole number(s) within ¾ of x. Is this a function?
No, this is not a function because if x =3D ½, there'll be = two different outputs 0 or 1.
Notation and Alg= ebra of Functions
Definition: A <= span lang=3DEN-US style=3D'font-size:156%;color:red;mso-fareast-language:ZH-CN'= >function f is a procedure that assigns each input x from a se= t D (the domain), exactly one output y in a set R (t= he range).  We write it as f(x) = =3D y,
where x = =3D input, y =3D output, or as f: D à R.
------=_NextPart_01C82E62.72341DF0 Content-Location: file:///C:/4D764A33/2NotationandAlgebraofFunctions_files/slide0143.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
Example A:
a. Input a number x, the output is (are)
whole number(s) within ¾ of x. Is this a function?
No, this is not a function because if x =3D ½, there'll be = two different outputs 0 or 1.
Notation and Alg= ebra of Functions
b. Input= a number x, the output is the largest whole = number less than or equal to  x. Is = this a function?
Definition: A <= span lang=3DEN-US style=3D'font-size:156%;color:red;mso-fareast-language:ZH-CN'= >function f is a procedure that assigns each input x from a se= t D (the domain), exactly one output y in a set R (t= he range).  We write it as f(x) = =3D y,
where x = =3D input, y =3D output, or as f: D à R.
------=_NextPart_01C82E62.72341DF0 Content-Location: file:///C:/4D764A33/2NotationandAlgebraofFunctions_files/slide0143_image001.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAIAgAAAAAAAAAIFjI+pywUAOw== ------=_NextPart_01C82E62.72341DF0 Content-Location: file:///C:/4D764A33/2NotationandAlgebraofFunctions_files/slide0144.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
Example A:
a. Input a number x, the output is (are)
whole number(s) within ¾ of x. Is this a function?
No, this is not a function because if x =3D ½, there'll be = two different outputs 0 or 1.
Notation and Alg= ebra of Functions
b. Input= a number x, the output is the largest whole = number less than or equal to  x. Is = this a function?
This is a function. It’s the grestest integer function = denoted as [x]. 
Definition: A <= span lang=3DEN-US style=3D'font-size:156%;color:red;mso-fareast-language:ZH-CN'= >function f is a procedure that assigns each input x from a se= t D (the domain), exactly one output y in a set R (t= he range).  We write it as f(x) = =3D y,
where x = =3D input, y =3D output, or as f: D à R.
------=_NextPart_01C82E62.72341DF0 Content-Location: file:///C:/4D764A33/2NotationandAlgebraofFunctions_files/slide0144_image002.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAIAgAAAAAAAAAIFjI+pywUAOw== ------=_NextPart_01C82E62.72341DF0 Content-Location: file:///C:/4D764A33/2NotationandAlgebraofFunctions_files/slide0145.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
Example A:
a. Input a number x, the output is (are)
whole number(s) within ¾ of x. Is this a function?
No, this is not a function because if x =3D ½, there'll be = two different outputs 0 or 1.
Notation and Alg= ebra of Functions
b. Input= a number x, the output is the largest whole = number less than or equal to  x. Is = this a function?
This is a function. It’s the grestest integer function = denoted as [x].  For instant, [3] =3D= 3, [3.12] =3D 3.
Definition: A <= span lang=3DEN-US style=3D'font-size:156%;color:red;mso-fareast-language:ZH-CN'= >function f is a procedure that assigns each input x from a se= t D (the domain), exactly one output y in a set R (t= he range).  We write it as f(x) = =3D y,
where x = =3D input, y =3D output, or as f: D à R.
------=_NextPart_01C82E62.72341DF0 Content-Location: file:///C:/4D764A33/2NotationandAlgebraofFunctions_files/slide0145_image003.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAIAgAAAAAAAAAIFjI+pywUAOw== ------=_NextPart_01C82E62.72341DF0 Content-Location: file:///C:/4D764A33/2NotationandAlgebraofFunctions_files/slide0133.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
Ther are many ways to define functions.=
Notation and Alg= ebra of Functions
------=_NextPart_01C82E62.72341DF0 Content-Location: file:///C:/4D764A33/2NotationandAlgebraofFunctions_files/slide0146.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
Ther are many ways to define functions.=
Functions= may be defined by written instructions
(as in ex= ample A, part b). 
Notation and Alg= ebra of Functions
------=_NextPart_01C82E62.72341DF0 Content-Location: file:///C:/4D764A33/2NotationandAlgebraofFunctions_files/slide0147.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
Ther are many ways to define functions.=
Functions= may be defined by written instructions
(as in ex= ample A, part b).  For the function [x],
the domai= n D =3D {all real numbers}, 
the range= R =3D {all whole numbers}.
Notation and Alg= ebra of Functions
------=_NextPart_01C82E62.72341DF0 Content-Location: file:///C:/4D764A33/2NotationandAlgebraofFunctions_files/slide0147_image004.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAIAgAAAAAAAAAIFjI+pywUAOw== ------=_NextPart_01C82E62.72341DF0 Content-Location: file:///C:/4D764A33/2NotationandAlgebraofFunctions_files/slide0148.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
Ther are many ways to define functions.=
Functions= may be defined by written instructions
(as in ex= ample A, part b).  For the function [x],
the domai= n D =3D {all real numbers}, 
the range= R =3D {all whole numbers}.
Functions may be given as table. 
Notation and Alg= ebra of Functions
------=_NextPart_01C82E62.72341DF0 Content-Location: file:///C:/4D764A33/2NotationandAlgebraofFunctions_files/slide0148_image005.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAIAgAAAAAAAAAIFjI+pywUAOw== ------=_NextPart_01C82E62.72341DF0 Content-Location: file:///C:/4D764A33/2NotationandAlgebraofFunctions_files/slide0149.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
Ther are many ways to define functions.=
Functions= may be defined by written instructions
(as in ex= ample A, part b).  For the function [x],
the domai= n D =3D {all real numbers}, 
the range= R =3D {all whole numbers}.
Functions may be given as table.  For example:
     2<= /span>
7
     4<= /span>
6
    -3
5
     3<= /span>
2
     4<= /span>
-1
y=3Df(x)
x
Notation and Alg= ebra of Functions
------=_NextPart_01C82E62.72341DF0 Content-Location: file:///C:/4D764A33/2NotationandAlgebraofFunctions_files/slide0149_image006.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhYQC4AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAABf ALcAgAAAAAAAAAL/jI+py+0Po5y02muA3rz7rwXgSJZmeZzqJq7ua6YwOtc2IN9eq/dx5uPwgsRO jjgsKo/BpBIJLDqfPmaV+rT2ptibVsft1r5eMTRsQ5td5PS6GkW+t/HmHFy/3tP57Z6vNhP4B9I2 RijYB4YIYyjI+OLYCMmmWEapIhmJmWnpxvkzuAmK4nlISqLJhpqK4foK+5DFOqK6Ioppm0lbaPrI u+M7CWwkPErMYryKnIx7y9w8C42j/Dyte+IMiR0zTa2d7c1Nc129Ww7ejS4NPZ4qbh6+vgSfTt4e r45v/z4vVc+OmbtaAOn5k3PQDrpYDBtiCIhsYKGC/xLq2QeRmMQP//z2bNxBESFGgyMrlhQpMN+9 lB0nWvTzclHMSycV1rzIMiOwj0ZC2sxJEqhJoSgjqux3E2ZSmUtpEv1p1KHUqRB08uIpxCfOqFZp YWWhVenTrRqPEpz5qWnasWK5BnU7FG7Rsi05hmXKFq9cqHS7svoa4q7TvWR3mnWp9lTiX4uHNT6W d3Dft3SpWr582C5axZHXEm47OW7ouYbrgtzMuDPnz3pH8y3tFxVgHII9uy58NfPpx8t4W/N9Drg8 4fpUp2YtGTZl5aKZk85tuidqx8apR8WMXWpsUrOjm+lee/Vt0M5fQ98OCvx0yMhtl8ftVbd04ivb i39P/vxy/c35PyeP7x1Y6/VWHXvjtYYfgv6ZByB6nKhHH1IFEmjfcZVlh2GGGm5oQAEAOw== ------=_NextPart_01C82E62.72341DF0 Content-Location: file:///C:/4D764A33/2NotationandAlgebraofFunctions_files/slide0149_image007.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAIAgAAAAAAAAAIFjI+pywUAOw== ------=_NextPart_01C82E62.72341DF0 Content-Location: file:///C:/4D764A33/2NotationandAlgebraofFunctions_files/slide0150.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
Ther are many ways to define functions.=
Functions= may be defined by written instructions
(as in ex= ample A, part b).  For the function [x],
the domai= n D =3D {all real numbers}, 
the range= R =3D {all whole numbers}.
Functions may be given as table.  For example:
     2<= /span>
7
     4<= /span>
6
    -3
5
     3<= /span>
2
     4<= /span>
-1
y=3Df(x)
x
For this function, the domain
D =3D {-1, 2, 5, 6, 7}, the range is
R =3D {4,= 3, -3, 2}.
Notation and Alg= ebra of Functions
------=_NextPart_01C82E62.72341DF0 Content-Location: file:///C:/4D764A33/2NotationandAlgebraofFunctions_files/slide0150_image008.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhYQC4AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAABf ALcAgAAAAAAAAAL/jI+py+0Po5y02muA3rz7rwXgSJZmeZzqJq7ua6YwOtc2IN9eq/dx5uPwgsRO jjgsKo/BpBIJLDqfPmaV+rT2ptibVsft1r5eMTRsQ5td5PS6GkW+t/HmHFy/3tP57Z6vNhP4B9I2 RijYB4YIYyjI+OLYCMmmWEapIhmJmWnpxvkzuAmK4nlISqLJhpqK4foK+5DFOqK6Ioppm0lbaPrI u+M7CWwkPErMYryKnIx7y9w8C42j/Dyte+IMiR0zTa2d7c1Nc129Ww7ejS4NPZ4qbh6+vgSfTt4e r45v/z4vVc+OmbtaAOn5k3PQDrpYDBtiCIhsYKGC/xLq2QeRmMQP//z2bNxBESFGgyMrlhQpMN+9 lB0nWvTzclHMSycV1rzIMiOwj0ZC2sxJEqhJoSgjqux3E2ZSmUtpEv1p1KHUqRB08uIpxCfOqFZp YWWhVenTrRqPEpz5qWnasWK5BnU7FG7Rsi05hmXKFq9cqHS7svoa4q7TvWR3mnWp9lTiX4uHNT6W d3Dft3SpWr582C5axZHXEm47OW7ouYbrgtzMuDPnz3pH8y3tFxVgHII9uy58NfPpx8t4W/N9Drg8 4fpUp2YtGTZl5aKZk85tuidqx8apR8WMXWpsUrOjm+lee/Vt0M5fQ98OCvx0yMhtl8ftVbd04ivb i39P/vxy/c35PyeP7x1Y6/VWHXvjtYYfgv6ZByB6nKhHH1IFEmjfcZVlh2GGGm5oQAEAOw== ------=_NextPart_01C82E62.72341DF0 Content-Location: file:///C:/4D764A33/2NotationandAlgebraofFunctions_files/slide0150_image009.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAIAgAAAAAAAAAIFjI+pywUAOw== ------=_NextPart_01C82E62.72341DF0 Content-Location: file:///C:/4D764A33/2NotationandAlgebraofFunctions_files/slide0151.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
Ther are many ways to define functions.=
Functions= may be defined by written instructions
(as in ex= ample A, part b).  For the function [x],
the domai= n D =3D {all real numbers}, 
the range= R =3D {all whole numbers}.
Functions may be given as table.  For example:
     2<= /span>
7
     4<= /span>
6
    -3
5
     3<= /span>
2
     4<= /span>
-1
y=3Df(x)
x
For this function, the domain
D =3D {-1, 2, 5, 6, 7}, the range is
R =3D {4,= 3, -3, 2}.
Specically, f(2) <= span lang=3DEN-US style=3D'font-size:156%;mso-fareast-language:ZH-CN'>=3D 3, f<= span lang=3DEN-US style=3D'font-size:156%;color:red;mso-fareast-language:ZH-CN'= >(6) =3D 4, et= c..
Notation and Alg= ebra of Functions
------=_NextPart_01C82E62.72341DF0 Content-Location: file:///C:/4D764A33/2NotationandAlgebraofFunctions_files/slide0151_image010.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhYQC4AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAABf ALcAgAAAAAAAAAL/jI+py+0Po5y02muA3rz7rwXgSJZmeZzqJq7ua6YwOtc2IN9eq/dx5uPwgsRO jjgsKo/BpBIJLDqfPmaV+rT2ptibVsft1r5eMTRsQ5td5PS6GkW+t/HmHFy/3tP57Z6vNhP4B9I2 RijYB4YIYyjI+OLYCMmmWEapIhmJmWnpxvkzuAmK4nlISqLJhpqK4foK+5DFOqK6Ioppm0lbaPrI u+M7CWwkPErMYryKnIx7y9w8C42j/Dyte+IMiR0zTa2d7c1Nc129Ww7ejS4NPZ4qbh6+vgSfTt4e r45v/z4vVc+OmbtaAOn5k3PQDrpYDBtiCIhsYKGC/xLq2QeRmMQP//z2bNxBESFGgyMrlhQpMN+9 lB0nWvTzclHMSycV1rzIMiOwj0ZC2sxJEqhJoSgjqux3E2ZSmUtpEv1p1KHUqRB08uIpxCfOqFZp YWWhVenTrRqPEpz5qWnasWK5BnU7FG7Rsi05hmXKFq9cqHS7svoa4q7TvWR3mnWp9lTiX4uHNT6W d3Dft3SpWr582C5axZHXEm47OW7ouYbrgtzMuDPnz3pH8y3tFxVgHII9uy58NfPpx8t4W/N9Drg8 4fpUp2YtGTZl5aKZk85tuidqx8apR8WMXWpsUrOjm+lee/Vt0M5fQ98OCvx0yMhtl8ftVbd04ivb i39P/vxy/c35PyeP7x1Y6/VWHXvjtYYfgv6ZByB6nKhHH1IFEmjfcZVlh2GGGm5oQAEAOw== ------=_NextPart_01C82E62.72341DF0 Content-Location: file:///C:/4D764A33/2NotationandAlgebraofFunctions_files/slide0151_image011.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAIAgAAAAAAAAAIFjI+pywUAOw== ------=_NextPart_01C82E62.72341DF0 Content-Location: file:///C:/4D764A33/2NotationandAlgebraofFunctions_files/slide0152.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
Ther are many ways to define functions.=
Functions= may be defined by written instructions
(as in ex= ample A, part b).  For the function [x],
the domai= n D =3D {all real numbers}, 
the range= R =3D {all whole numbers}.
Functions may be given as table.  For example:
     2<= /span>
7
     4<= /span>
6
    -3
5
     3<= /span>
2
     4<= /span>
-1
y=3Df(x)
x
For this function, the domain
D =3D {-1, 2, 5, 6, 7}, the range is
R =3D {4,= 3, -3, 2}.
Specically, f(2) <= span lang=3DEN-US style=3D'font-size:156%;mso-fareast-language:ZH-CN'>=3D 3, f<= span lang=3DEN-US style=3D'font-size:156%;color:red;mso-fareast-language:ZH-CN'= >(6) =3D 4, et= c..
Note that we have both f(-2) = =3D f(6) =3D 4.
Notation and Alg= ebra of Functions
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Ther are many ways to define functions.=
Functions= may be defined by written instructions
(as in ex= ample A, part b).  For the function [x],
the domai= n D =3D {all real numbers}, 
the range= R =3D {all whole numbers}.
Functions may be given as table.  For example:
     2<= /span>
7
     4<= /span>
6
    -3
5
     3<= /span>
2
     4<= /span>
-1
y=3Df(x)
x
For this function, the domain
D =3D {-1, 2, 5, 6, 7}, the range is
R =3D {4,= 3, -3, 2}.
Specically, f(2) <= span lang=3DEN-US style=3D'font-size:156%;mso-fareast-language:ZH-CN'>=3D 3, f<= span lang=3DEN-US style=3D'font-size:156%;color:red;mso-fareast-language:ZH-CN'= >(6) =3D 4, et= c..
Note that we have both f(-2) = =3D f(6) =3D 4.
Its OK for different i= nputs to have the same output for functions.
Notation and Alg= ebra of Functions
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Functions may be given graphically:
Notation and Alg= ebra of Functions
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Functions may be given graphically:
For instance, Nominal Price(1975) » $0.50
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Notation and Alg= ebra of Functions
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Functions may be given graphically:
Domain =3D {year 1918 à 2005}
For instance, Nominal Price(1975) » $0.50
Notation and Alg= ebra of Functions
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Functions may be given graphically:
Domain =3D {year 1918 à 2005}
Range (Nominal Price) =3D {$0.20à$2.51}
For instance, Nominal Price(1975) » $1.00
Notation and Alg= ebra of Functions
<= span lang=3DZH-CN style=3D'mso-ansi-language:EN-US;mso-special-format:lastCR; display:none'>
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Notation and Alg= ebra of Functions
Functions may be given graphically:
Inflation Adjusted Price(1975) » $1.85
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file:///C:/4D764A33/2NotationandAlgebraofFunctions_files/slide0157.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
Notation and Alg= ebra of Functions
Functions may be given graphically:
Domain =3D {year 1918 à 2005}
Inflation Adjusted Price(1975) » $1.85
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Notation and Alg= ebra of Functions
Functions may be given graphically:
Domain =3D {year 1918 à 2005}
Range (Inflation Adjusted Price) =3D {$1.25à= $3.50}
Inflation Adjusted Price(1975) » $1.85
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