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Notation and Alg= ebra of Functions
Example C.
Given f(x) =3D x – 2x + 3, simplify the following.
a. f (2a) =3D (2a)2 – 2(2a)<= span lang=3DEN-US style=3D'font-size:156%;color:#003300;mso-fareast-language:ZH= -CN'> + 3
        = ;      =3D  4a2 – 4a + 3
b. f (a+b) =3D (a+b)2 – 2(a+b)= + 3
        = ;       =3D a + 2ab + b2 – 2a – 2b + 3
Graphs of Functions=
If we have a function = y =3D f(x), we may plot the points (x, y) that satisfy the function in the x&y coordinate system. The result is the graph of the = function.=  
We may use expressions as inputs for functions= .
Reminder: (a ± b)2 =3D a ± 2ab + b
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Notation and Alg= ebra of Functions
Given the function = ;
y =3D f(x) =3D x + 1, = make a table a= nd graph it.
------=_NextPart_01C9689C.78F4B620 Content-Location: file:///C:/4D764A33/6NotationofFunctions_files/slide0207.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
Notation and Alg= ebra of Functions
Given the function = ;
y =3D f(x) =3D x + 1, = make a table a= nd graph it.
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Notation and Alg= ebra of Functions
Given the function = ;
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Notation and Alg= ebra of Functions
Given the function = ;
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Notation and Alg= ebra of Functions
Given the function = ;
y =3D f(x) =3D x + 1, = make a table a= nd graph it.
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y
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y =3D x + 1
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Notation and Alg= ebra of Functions
Given the function = ;
y =3D f(x) =3D x + 1, = make a table a= nd graph it.
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y =3D x + 1
In general given an equation of x and y, we may = plot all = the ordered pair (x, y)’s that satisfy the equation = in the x&y coordinate system.
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Notation and Alg= ebra of Functions
Given the function = ;
y =3D f(x) =3D x + 1, = make a table a= nd graph it.
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y
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y =3D x + 1
In general given an equation of x and y, we may = plot all = the ordered pair (x, y)’s that satisfy the equation = in the x&y coordinate system. The result is the graph of the equation.
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Notation and Alg= ebra of Functions
The equation x =3D y2 is not a function because if =
x =3D 4, there are two= outputs y that satisfy 4 =3D y2, =
------=_NextPart_01C9689C.78F4B620 Content-Location: file:///C:/4D764A33/6NotationofFunctions_files/slide0218.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
Notation and Alg= ebra of Functions
The equation x =3D y2 is not a function because if =
x =3D 4, there are two= outputs y that satisfy 4 =3D y2,
y =3D 2 and y =3D R= 11;2.
------=_NextPart_01C9689C.78F4B620 Content-Location: file:///C:/4D764A33/6NotationofFunctions_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
The equation x =3D y2 is not a function because if =
x =3D 4, there are two= outputs y that satisfy 4 =3D y2,
y =3D 2 and y =3D R= 11;2. So this is not a function.
------=_NextPart_01C9689C.78F4B620 Content-Location: file:///C:/4D764A33/6NotationofFunctions_files/slide0220.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
Notation and Alg= ebra of Functions
The equation x =3D y2 is not a function because if =
x =3D 4, there are two= outputs y that satisfy 4 =3D y2,
y =3D 2 and y =3D R= 11;2. So this is not a function.
We make a table for it’s graph.
y
4
4
1
1
0
x
------=_NextPart_01C9689C.78F4B620 Content-Location: file:///C:/4D764A33/6NotationofFunctions_files/slide0220_image023.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhqAA4AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACm ADYAgAAAAAAAAAL/jI+py+0Po5y0ToCz3nyHDobYJ5aeiQJkKq5seLytDNO1zbn4uOdGr9HthENg hmhDJn9GVdP5VMqkL2qVabSmtNuoN4oFck1j8jf7VIV75dlZ/Bavi822G11Px/L8PtwvZwci2EFY GEdXN4djeILI+Mi4mBRJCZiYNUnTGKRXOaWnCXqJyfY5JVp1qkpqqch5tMoCy9O6RAuFZzuqm4n7 67nL+iqLgpv71xtYbMZcAgwGrWwqPBsqnTwNWT2bas3dBd5MLP4crO3q63yHvrneYhEvP09fbz// PnieTU3ezvtvGD823sIFFNhv4JCCxvLpK8duGcQbEx8e7IYtocJ0khoJZizVcdvFLQzHbeQIUqS6 iocc5nDp46PKkwBDSpKJMmXOnbdgdvIZi2WhkuaEvjT60x9NhDbdIQ1CNKJOnjWnbopKcSnTmVZR 4XT6tNZIY1gtav02tqjSplW7oj1L8mtbrnSpegUqNq3Uuj3DIoNrku1dv8fkbr2J96/gt4sxJi78 +FrkyXrh3buMObPmzQUAADs= ------=_NextPart_01C9689C.78F4B620 Content-Location: file:///C:/4D764A33/6NotationofFunctions_files/slide0221.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
Notation and Alg= ebra of Functions
The equation x =3D y2 is not a function because if =
x =3D 4, there are two= outputs y that satisfy 4 =3D y2,
y =3D 2 and y =3D R= 11;2. So this is not a function.
We make a table for it’s graph.
0
y
4
4
1
1
0
x
------=_NextPart_01C9689C.78F4B620 Content-Location: file:///C:/4D764A33/6NotationofFunctions_files/slide0221_image024.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhqAA4AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACm ADYAgAAAAAAAAAL/jI+py+0Po5y0ToCz3nyHDobYJ5aeiQJkKq5seLytDNO1zbn4uOdGr9HthENg hmhDJn9GVdP5VMqkL2qVabSmtNuoN4oFck1j8jf7VIV75dlZ/Bavi822G11Px/L8PtwvZwci2EFY GEdXN4djeILI+Mi4mBRJCZiYNUnTGKRXOaWnCXqJyfY5JVp1qkpqqch5tMoCy9O6RAuFZzuqm4n7 67nL+iqLgpv71xtYbMZcAgwGrWwqPBsqnTwNWT2bas3dBd5MLP4crO3q63yHvrneYhEvP09fbz// PnieTU3ezvtvGD823sIFFNhv4JCCxvLpK8duGcQbEx8e7IYtocJ0khoJZizVcdvFLQzHbeQIUqS6 iocc5nDp46PKkwBDSpKJMmXOnbdgdvIZi2WhkuaEvjT60x9NhDbdIQ1CNKJOnjWnbopKcSnTmVZR 4XT6tNZIY1gtav02tqjSplW7oj1L8mtbrnSpegUqNq3Uuj3DIoNrku1dv8fkbr2J96/gt4sxJi78 +FrkyXrh3buMObPmzQUAADs= ------=_NextPart_01C9689C.78F4B620 Content-Location: file:///C:/4D764A33/6NotationofFunctions_files/slide0222.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
Notation and Alg= ebra of Functions
The equation x =3D y2 is not a function because if =
x =3D 4, there are two= outputs y that satisfy 4 =3D y2,
y =3D 2 and y =3D R= 11;2. So this is not a function.
We make a table for it’s graph.
-1
1
0
y
4
4
1
1
0
x
------=_NextPart_01C9689C.78F4B620 Content-Location: file:///C:/4D764A33/6NotationofFunctions_files/slide0222_image025.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhqAA4AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACm ADYAgAAAAAAAAAL/jI+py+0Po5y0ToCz3nyHDobYJ5aeiQJkKq5seLytDNO1zbn4uOdGr9HthENg hmhDJn9GVdP5VMqkL2qVabSmtNuoN4oFck1j8jf7VIV75dlZ/Bavi822G11Px/L8PtwvZwci2EFY GEdXN4djeILI+Mi4mBRJCZiYNUnTGKRXOaWnCXqJyfY5JVp1qkpqqch5tMoCy9O6RAuFZzuqm4n7 67nL+iqLgpv71xtYbMZcAgwGrWwqPBsqnTwNWT2bas3dBd5MLP4crO3q63yHvrneYhEvP09fbz// PnieTU3ezvtvGD823sIFFNhv4JCCxvLpK8duGcQbEx8e7IYtocJ0khoJZizVcdvFLQzHbeQIUqS6 iocc5nDp46PKkwBDSpKJMmXOnbdgdvIZi2WhkuaEvjT60x9NhDbdIQ1CNKJOnjWnbopKcSnTmVZR 4XT6tNZIY1gtav02tqjSplW7oj1L8mtbrnSpegUqNq3Uuj3DIoNrku1dv8fkbr2J96/gt4sxJi78 +FrkyXrh3buMObPmzQUAADs= ------=_NextPart_01C9689C.78F4B620 Content-Location: file:///C:/4D764A33/6NotationofFunctions_files/slide0223.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
Notation and Alg= ebra of Functions
The equation x =3D y2 is not a function because if =
x =3D 4, there are two= outputs y that satisfy 4 =3D y2,
y =3D 2 and y =3D R= 11;2. So this is not a function.
We make a table for it’s graph.
-2
2
-1
1
0
y
4
4
1
1
0
x
------=_NextPart_01C9689C.78F4B620 Content-Location: file:///C:/4D764A33/6NotationofFunctions_files/slide0223_image026.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhqAA4AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACm ADYAgAAAAAAAAAL/jI+py+0Po5y0ToCz3nyHDobYJ5aeiQJkKq5seLytDNO1zbn4uOdGr9HthENg hmhDJn9GVdP5VMqkL2qVabSmtNuoN4oFck1j8jf7VIV75dlZ/Bavi822G11Px/L8PtwvZwci2EFY GEdXN4djeILI+Mi4mBRJCZiYNUnTGKRXOaWnCXqJyfY5JVp1qkpqqch5tMoCy9O6RAuFZzuqm4n7 67nL+iqLgpv71xtYbMZcAgwGrWwqPBsqnTwNWT2bas3dBd5MLP4crO3q63yHvrneYhEvP09fbz// PnieTU3ezvtvGD823sIFFNhv4JCCxvLpK8duGcQbEx8e7IYtocJ0khoJZizVcdvFLQzHbeQIUqS6 iocc5nDp46PKkwBDSpKJMmXOnbdgdvIZi2WhkuaEvjT60x9NhDbdIQ1CNKJOnjWnbopKcSnTmVZR 4XT6tNZIY1gtav02tqjSplW7oj1L8mtbrnSpegUqNq3Uuj3DIoNrku1dv8fkbr2J96/gt4sxJi78 +FrkyXrh3buMObPmzQUAADs= ------=_NextPart_01C9689C.78F4B620 Content-Location: file:///C:/4D764A33/6NotationofFunctions_files/slide0217.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
Notation and Alg= ebra of Functions
The equation x =3D y2 is not a function because if =
x =3D 4, there are two= outputs y that satisfy 4 =3D y2,
y =3D 2 and y =3D R= 11;2. So this is not a function.
We make a table for it’s graph.
-2
2
-1
1
0
y
4
4
1
1
0
x
=
-2
2
-1
1
0
y
4
4
1
1
0
x
x =3D y2 = ;
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Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
Notation and Alg= ebra of Functions
The equation x =3D y2 is not a function because if =
x =3D 4, there are two= outputs y that satisfy 4 =3D y2,
y =3D 2 and y =3D R= 11;2. So this is not a function.
We make a table for it’s graph.
-2
2
-1
1
0
y
4
4
1
1
0
x
-2
2
-1
1
0
y
4
4
1
1
0
x
x =3D y2 = ;
Note that because (4, 2) and (4, –2) are on the graph = if we dra= w a vertical line at x =3D 4, it intersects the = graph at = two points.
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text/html; charset="us-ascii" Notation and Algebra of Functions
Notation and Alg= ebra of Functions
The equation x =3D y2 is not a function because if =
x =3D 4, there are two= outputs y that satisfy 4 =3D y2,
y =3D 2 and y =3D R= 11;2. So this is not a function.
We make a table for it’s graph.
-2
2
-1
1
0
y
4
4
1
1
0
x
-2
2
-1
1
0
y
4
4
1
1
0
x
x =3D y2 = ;
Note that because (4, 2) and (4, –2) are on the graph = if we dra= w a vertical line at x =3D 4, it intersects the = graph at = two points.
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Algebra of Functions
Notation and Alg= ebra of Functions
Since for functions each input x has exactly one = output, t= herefore each vertical line can only intersect it’s g= raph at exactly one location (eg. y =3D x + 1).
------=_NextPart_01C9689C.78F4B620 Content-Location: file:///C:/4D764A33/6NotationofFunctions_files/slide0226.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
Notation and Alg= ebra of Functions
Since for functions each input x has exactly one = output, t= herefore each vertical line can only intersect it’s g= raph at exactly one location (eg. y =3D x + 1).
y =3D x + 1
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Notation and Alg= ebra of Functions
Since for functions each input x has exactly one = output, t= herefore each vertical line can only intersect it’s g= raph at exactly one location (eg. y =3D x + 1).
y =3D x + 1
However, if any vertical line intersects a graph at two or = more poin= ts, i.e. there are two or more outputs y associated = to one in= put x (eg. x =3D y2),
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Notation and Alg= ebra of Functions
Since for functions each input x has exactly one = output, t= herefore each vertical line can only intersect it’s g= raph at exactly one location (eg. y =3D x + 1).
y =3D x + 1
However, if any vertical line intersects a graph at two or = more poin= ts, i.e. there are two or more outputs y associated = to one in= put x (eg. x =3D y2), then the = graph must not be the graph of a function.
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