User:Oneismany/Sandbox

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Oneismany's Sandbox

LaTeX testing

Displaying a formula

y = x2

f(x) = x2

y2 = x

y^2 = x \Leftrightarrow |y| = \sqrt{x}

y = x^2 \Leftrightarrow \sqrt{y} = |x|

NOT A FUNCTION: y^2 = x \Leftrightarrow |y| = \sqrt{x} \Leftrightarrow y = \pm\sqrt{x}

FUNCTION: y = x^2 \Leftrightarrow \sqrt{y} = |x| \Leftrightarrow \pm\sqrt{y} = x

f(x) = \sqrt{x}

f(x) = -\sqrt{x}

f(x) = \pm\sqrt{x}

NOT A FUNCTION: f(x) = \pm\sqrt{x}

| f(x) | = x

f(x) = \pm{x}

| y | = x

y = \pm{x}

y = | x |

\pm{y} = x

|y| = x \Leftrightarrow y = \pm{x}

y = |x| \Leftrightarrow \pm{y} = x

NOT A FUNCTION: |y| = x \Leftrightarrow y = \pm{x} [2 DIFFERENT y's, SAME x]

FUNCTION: y = |x| \Leftrightarrow \pm{y} = x [2 DIFFERENT x's, SAME y]

f(x) = 2x + 3 \Leftrightarrow y = 2x + 3

Inverse: x = 2y + 3 \Leftrightarrow y = \frac{1}{2}x + \frac{3}{2} \Leftrightarrow f(x) = \frac{1}{2}x + \frac{3}{2}

x^2 + y^2 = 25 \Leftrightarrow y = \pm\sqrt{25 - x^2}

y = \sqrt{25 - x^2}

y = -\sqrt{25 - x^2}

|a - b| = d \Leftrightarrow (a - b) = \pm{d}

f(x) = \begin{cases} \frac{x^2-9}{x+3} \ \ \ \ for x \ne -3 \\ 5 \ \ \ \ \ \ \ \ for x = -3 \end{cases}

2

f(2 − x) = 2(2 − x)2 + 6(2 − x) + 15 = 2(4 − 4x + x2) + 12 − 6x + 15

= 8 − 8x + 2x2 + 12 − 6x + 15 = 2x2 − 14x + 35

f(x) = 2x2 − 14x + 35

slope = m = \frac{-3-(-19)}{-12-(-20)} = \frac{16}{2} = 2

(y - (-3)) = 2(x - (-12)) \Leftrightarrow y + 3 = 2x + 24

\Leftrightarrow y = 2x + 21

f(x) = 2x + 21

g(f(x)) = (\sqrt{x-7})^2 + 15 = |x - 7| + 15

| x − 7 | + 15

g(f( − 8)) = | − 8 − 7 | + 15 = | − 15 | + 15 = 30

f(x) = x2 − x4

f( − x) = ( − x)2 − ( − x)4

\ \ \ = x^2 - x^4

\ \ \ = f(x)

f(x) = x3

f(x) = \sqrt[3]{x}

f(x) = | x |

f(x) = x

f(x) = \frac{1}{x}

f( − x) = f(x)

f( − x) = − f(x)

EVEN FUNCTIONS

\begin{align} f(x) & = x^2 - x^4 \\ f(-x)& = (-x)^2 - (-x)^4 \\ & = x^2 - x^4 \\ & = f(x) \end{align}

\begin{align} f(x) & = \frac{1}{x^6} + x^8 - 4 \\ f(-x)& = \frac{1}{(-x)^6} + (-x)^8 - 4 \\ & = \frac{1}{x^6} + x^8 - 4 \\ & = f(x) \end{align}


f(x) = x2 − x4

f(x) = \frac{1}{x^6} + x^8 - 4

ODD FUNCTIONS

\begin{align} f(x) & = x^3 + x^5 \\ f(-x)& = (-x)^3 + (-x)^5 \\ & = -x^3 + (-x^5) \\ & = -(x^3 + x^5) \\ & = -f(x) \end{align}

\begin{align} f(x) & = \sqrt[3]{x} + \sqrt[5]{x} \\ f(-x)& = \sqrt[3]{-x} + \sqrt[5]{-x} \\ & = -\sqrt[3]{x} + (-\sqrt[5]{x}) \\ & = -(\sqrt[3]{x} + \sqrt[5]{x}) \\ & = -f(x) \end{align}


f(x) = x3 + x5

f(x) = \sqrt[3]{x} + \sqrt[5]{x}

Force = G \frac{m_1m_2}{d^2}


Tables side by side / tables as cells in a larger table

SSAA Winter World
Savannah, GA
January 6-7
.584 Team Avg - 15.3 Runs/Game
Player RS BB/H AB HR AVG
Josh B
15
23
40
0
.575
Brandon
12
19
31
0
.613
Jeff
17
25
36
2
.694
Chad
17
22
35
1
.629
JP
9
21
34
0
.618
Shane
13
18
37
0
.486
Scott
19
20
36
3
.556
Ron
11
20
34
0
.588
Brian
12
18
33
3
.545
Jason
16
25
37
1
.676
Mike
12
16
36
0
.444
ISA Miken Winter Warm Up
Lexington, SC
February 10-11
.635 Team Avg - 15.4 Runs/Game
Player RS BB/H AB HR AVG
Josh B
-
36
45
0
.800
Josh E
-
18
34
0
.529
Brandon
-
18
34
0
.529
Jeff
-
28
38
3
.737
Chad
-
25
39
0
.641
JP
-
22
37
1
.595
Shane
-
31
43
2
.721
Scott
-
24
39
2
.615
Brian
-
22
38
2
.579
Jason
-
26
45
1
.578
Mike
-
21
35
0
.600