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------=_NextPart_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_files/master06.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_files/preview.wmf Content-Transfer-Encoding: base64 Content-Type: image/x-wmf AQAJAAAD4gAAAAUAOQAAAAAABAAAAAMBCAAFAAAACwIAAAAABQAAAAwC7wLpAwMAAAAeAAcAAAD8 AgAA////AAAABAAAAC0BAAAIAAAA+gIFAAAAAAD///8ABAAAAC0BAQAMAAAAQAkhAPAAAAAAAAAA 7wLpA/////8IAAAA+gIAAAAAAAAAAAAABAAAAC0BAgAEAAAALQEAAAQAAAAnAf//HAAAAPsC1P8A AAAAAACQAQAAAAAAQAAAQXJpYWwAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAEAAAALQEDAAQA AAAuARgABAAAAAIBAQAFAAAACQIAAAACOQAAADIKNAC5ACEAAABOb3RhdGlvbiBhbmQgQWxnZWJy YSBvZiBGdW5jdGlvbnMgIAAYAA0AGAAMAAsAGAAZAAwAGQAZABgADQAdAAoAGQAYABkADwAYAA0A GAANAAwAGwAZABgAFgAMAAsAGQAYABYABAAAAC4BAAAcAAAA+wIQAAcAAAAAALwCAAAAAAECAiJT eXN0ZW0AAAAAAAAYAAAALJ4TAAEAAADkBAAAAAAAAAQAAAAtAQQABAAAAPABAwADAAAAAAA= ------=_NextPart_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_files/pres.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_files/slide0081.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
Notation and Alg= ebra of Functions
------=_NextPart_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_files/slide0219.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). 
------=_NextPart_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_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_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_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_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_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_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_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_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_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_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_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 greatest 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_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_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 greatest 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_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_files/slide0133.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
There are many ways to define functions.
Notation and Alg= ebra of Functions
------=_NextPart_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_files/slide0146.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
There 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_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_files/slide0147.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
There 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_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_files/slide0148.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
There 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_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_files/slide0149.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
There 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_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_files/slide0149_image001.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_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_files/slide0150.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
There 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_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_files/slide0150_image002.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_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_files/slide0151.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
There 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}.
Specifically, 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_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_files/slide0151_image003.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_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_files/slide0152.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
There 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}.
Note that we have both f(-1) = =3D f(6) =3D 4.
Notation and Alg= ebra of Functions
Specifically, 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..
------=_NextPart_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_files/slide0152_image004.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_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_files/slide0153.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
There 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}.
Note that we have both f(-1) = =3D f(6) =3D 4.
Its OK for different i= nputs to have the same output for functions.
Notation and Alg= ebra of Functions
Specifically, 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..
------=_NextPart_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_files/slide0153_image005.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_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_files/slide0134.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
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
<= span lang=3DEN-US style=3D'mso-fareast-language:ZH-CN;mso-special-format:lastCR; display:none'>
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
<= span lang=3DEN-US style=3D'mso-fareast-language:ZH-CN;mso-special-format:lastCR; display:none'>
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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=3DEN-US style=3D'mso-fareast-language:ZH-CN;mso-special-format:lastCR; display:none'>
------=_NextPart_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_files/slide0135.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:
Inflation Adjusted Price(1975) » $1.85
<= span lang=3DEN-US style=3D'mso-fareast-language:ZH-CN;mso-special-format:lastCR; display:none'>
------=_NextPart_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_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
<= span lang=3DEN-US style=3D'mso-fareast-language:ZH-CN;mso-special-format:lastCR; display:none'>
------=_NextPart_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_files/slide0158.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}
Range (Inflation Adjusted Price) =3D {$1.25à= $3.50}
Inflation Adjusted Price(1975) » $1.85
<= span lang=3DEN-US style=3D'mso-fareast-language:ZH-CN;mso-special-format:lastCR; display:none'>
------=_NextPart_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_files/slide0090.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
Notation and Alg= ebra of Functions
Most functions are given by mathematics formulas.
------=_NextPart_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_files/slide0159.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
For example,  =     
        = ;  f(X) =3D X2 – 2X + 3 =3D y
Notation and Alg= ebra of Functions
Most functions are given by mathematics formulas.
------=_NextPart_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_files/slide0160.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
For example,  =     
        = ;  f(X) =3D X2 – 2X + 3 =3D y
name of        = ;            &n= bsp;     
the function
Notation and Alg= ebra of Functions
Most functions are given by mathematics formulas.
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For example,  =     
        = ;  f(X) =3D X2 – 2X + 3 =3D y
name of        = ;            &n= bsp;        
the function
Notation and Alg= ebra of Functions
Most functions are given by mathematics formulas.
input box
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For example,  =     
        = ;  f(X) =3D X2 – 2X + 3 =3D y
name of        = ;            &n= bsp;         actual formula
the function
Notation and Alg= ebra of Functions
Most functions are given by mathematics formulas.
input box
------=_NextPart_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_files/slide0163.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
For example,  =     
        = ;  f(X) =3D X2 – 2X + 3 =3D y
name of        = ;            &n= bsp;         actual formula
the function
The outpu= t
Notation and Alg= ebra of Functions
Most functions are given by mathematics formulas.
input box
------=_NextPart_01C957BC.67F8B330 Content-Location: file:///C:/4D764A33/1FunctionsandOperationsofFunctions_files/slide0164.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
For example,  =     
        = ;  f(X) =3D X2 – 2X + 3 =3D y
name of        = ;            &n= bsp;         actual formula
the function
The outpu= t
Notation and Alg= ebra of Functions
Most functions are given by mathematics formulas.
input box
The input box holds the input for the formula.=
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For example,  =     
        = ;  f(X) =3D X2 – 2X + 3 =3D y
name of        = ;            &n= bsp;         actual formula
the function
The outpu= t
Notation and Alg= ebra of Functions
Most functions are given by mathematics formulas.
input box
The input= box holds the input for the formula.
Hence f (2) means to replace x by (2) in the formula, = so f(2) <= span lang=3DEN-US style=3D'font-size:156%;color:black;mso-fareast-language:ZH-C= N'>=3D <= span lang=3DEN-US style=3D'font-size:156%;color:#0000CC;mso-fareast-language:ZH= -CN'>(2)2 – 2(2) <= span lang=3DEN-US style=3D'font-size:156%;color:red;mso-fareast-language:ZH-CN'= >+ 3 =3D 3 =3D y.
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For example,  =     
        = ;  f(X) =3D X2 – 2X + 3 =3D y
name of        = ;            &n= bsp;         actual formula
the function
The outpu= t
Notation and Alg= ebra of Functions
Most functions are given by mathematics formulas.
input box
The input= box holds the input for the formula.
Hence f (2) means to replace x by (2) in the formula, = so f(2) <= span lang=3DEN-US style=3D'font-size:156%;color:black;mso-fareast-language:ZH-C= N'>=3D <= span lang=3DEN-US style=3D'font-size:156%;color:#0000CC;mso-fareast-language:ZH= -CN'>(2)2 – 2(2) <= span lang=3DEN-US style=3D'font-size:156%;color:red;mso-fareast-language:ZH-CN'= >+ 3 =3D 3 =3D y.
Real functions<= /span> are func= tions whose domain and range are subsets of real nu= mbers (no complex numbers).
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Domain of Functions
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Domain of Functions
There are = two main things to consider when determinin= g the domains of real functions.
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Domain of Functions
There are = two main things to consider when determinin= g the domains of real functions.
1. The denominators can't be 0
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Domain of Functions
There are = two main things to consider when determinin= g the domains of real functions.
1. The denominators can't be 0
2. The radicand of even roots can't be negative.
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Domain of Functions
There are = two main things to consider when determinin= g the domains of real functions.
1. The denominators can't be 0
2. The radicand of even roots can't be negative.
Example B: Find the domain of the following functions. <= /div>
a. f(x) =3D 1/(2x + 6)
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Domain of Functions
There are = two main things to consider when determinin= g the domains of real functions.
1. The denominators can't be 0
2. The radicand of even roots can't be negative.
Example B: Find the domain of the following functions. <= /div>
a. f(x) =3D 1/(2x + 6)
The denominator can’t be 0 i.e. 2x + 6 =3D 0
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Domain of Functions
There are = two main things to consider when determinin= g the domains of real functions.
1. The denominators can't be 0
2. The radicand of even roots can't be negative.
Example B: Find the domain of the following functions. <= /div>
a. f(x) =3D 1/(2x + 6)
The denominator can’t be 0 i.e. 2x + 6 =3D 0 à x =3D -3
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Domain of Functions
There are = two main things to consider when determinin= g the domains of real functions.
1. The denominators can't be 0
2. The radicand of even roots can't be negative.
Example B: Find the domain of the following functions. <= /div>
a. f(x) =3D 1/(2x + 6)
The denominator can’t be 0 i.e. 2x + 6 =3D 0 à x =3D -3
So the domain =3D {all numbers except x =3D -3}<= /i>.
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Domain of Functions
There are = two main things to consider when determinin= g the domains of real functions.
1. The denominators can't be 0
2. The radicand of even roots can't be negative.
Example B: Find the domain of the following functions. <= /div>
a. f(x) =3D 1/(2x + 6)
The denominator can’t be 0 i.e. 2x + 6 =3D 0 à x =3D -3
So the domain =3D {all numbers except x =3D -3}<= /i>.
b. f (X) =3D Ö 2x + 6
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Domain of Functions
There are = two main things to consider when determinin= g the domains of real functions.
1. The denominators can't be 0
2. The radicand of even roots can't be negative.
Example B: Find the domain of the following functions. <= /div>
a. f(x) =3D 1/(2x + 6)
The denominator can’t be 0 i.e. 2x + 6 =3D 0 à x =3D -3
So the domain =3D {all numbers except x =3D -3}<= /i>.
b. f (X) =3D Ö 2x + 6
we must have square root of nonnegative numbers.
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Domain of Functions
There are = two main things to consider when determinin= g the domains of real functions.
1. The denominators can't be 0
2. The radicand of even roots can't be negative.
Example B: Find the domain of the following functions. <= /div>
a. f(x) =3D 1/(2x + 6)
The denominator can’t be 0 i.e. 2x + 6 =3D 0 à x =3D -3
So the domain =3D {all numbers except x =3D -3}<= /i>.
b. f (X) =3D Ö 2x + 6
we must have square root of nonnegative numbers. <= span lang=3DEN-US style=3D'font-size:156%;color:#003300;mso-fareast-language:ZH= -CN'>Hence 2x + 6 > 0 &agrav= e; x > -3
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Domain of Functions
There are = two main things to consider when determinin= g the domains of real functions.
1. The denominators can't be 0
2. The radicand of even roots can't be negative.
Example B: Find the domain of the following functions. <= /div>
a. f(x) =3D 1/(2x + 6)
The denominator can’t be 0 i.e. 2x + 6 =3D 0 à x =3D -3
So the domain =3D {all numbers except x =3D -3}<= /i>.
b. f (X) =3D Ö 2x + 6
we must have square root of nonnegative numbers. <= span lang=3DEN-US style=3D'font-size:156%;color:#003300;mso-fareast-language:ZH= -CN'>Hence 2x + 6 > 0 &agrav= e; x > -3
So the domain =3D {all numbers x > -3}
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Domain of Functions
c. Find the domain of f(X) =3D&nbs= p;
   1
Ö
x + 6
 – 1
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Domain of Functions
c. Find the domain of f(X) =3D&nbs= p;
We must have square root of nonnegative numbers.
   1
Ö
x + 6
 – 1
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Domain of Functions
c. Find the domain of f(X) =3D&nbs= p;
We must have square root of nonnegative numbers.
   1
Ö
x + 6
 – 1
Hence x must satisfy the inequality
   1
x + 6
 – 1
>
0
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Domain of Functions
c. Find the domain of f(X) =3D&nbs= p;
We must have square root of nonnegative numbers.
   1
Ö
x + 6
 – 1
Hence x must satisfy the inequality
   1
x + 6
 – 1
>
0
This is a sign-chart problem. 
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Domain of Functions
c. Find the domain of f(X) =3D&nbs= p;
We must have square root of nonnegative numbers.
   1
Ö
x + 6
 – 1
Hence x must satisfy the inequality
   1
x + 6
 – 1
>
0
This is a sign-chart problem.  Put it in factored form:
 -x – 5
x + 6
>
0
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Domain of Functions