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4. Without application in the world, the value of knowledge is greatly diminished. Consider the claim with respect to two areas of knowledge.

sofia.callamand
Algebra and Function

1 Transformations

1.1 y = f(x) + a

1.1.1 Translation ( 0 , a )

1.1.1.1 Moves the graph up a units

1.2 y = f(x - a)

1.2.1 Translation ( a , 0 )

1.2.1.1 Subtracting a from
x shifts the graph
to the right

1.3 y = -f(x) is a
reflection in
the x-axis

1.3.1 y = f(-x) is a
reflection in
the y-axis

1.4 y = af(x) is a
stretch in the y
direction by a

1.4.1 y = f(ax) is a
stretch in the x
direction be
a^-1

1.5 For y = f(|x|)
for x > 0 and
reflects in y to
the right

1.5.1 For y = |f(x)| for y
< 0 reflected in
the line of the
dotted x-axis

2 Functions

2.1 A function is defined by:

2.1.1 A rule
connecting
the range and
domain sets

2.1.2 For each
member of the
domain, there
is only one
range value

2.2 A Function y = f(x)

2.2.1 One to One:
One X value
maps to one
Y value

2.2.2 Many to One:
More than
one value of X
maps to one
value of Y

2.3 Composite Funtion

2.3.1 fg(x) = f(g(x))
Put g into f

2.3.1.1 The output
of g becomes
the input of f

2.3.2 Can only be
formed in the
example of fg(x).
When the range
of g is in the
domain of f

2.4 Inverse Function

2.4.1 f^-1(x)

2.4.1.1 These can only
exist when f(x)
is a one to one
mapping

2.4.2 The range of f
is the domain
of f^-1 and vice
versa

2.4.2.1 The graph
y=f(x) is the
reflection of
y=f^-1(x)

2.4.3 To turn f(x) into
f^-1(x). Replace
the x with y's
and vice versa
then make y the
subject.

3 Modulus Function

3.1 |x| = x if x > 0

3.1.1 |x| = -x if x < 0

3.2 |x| < a = -a < x < a

3.2.1 |x| > a = x < -a or x > a

3.3 |x -a | = x - a for x >= a

3.3.1 |x - a| = -(x - a) = a - x for x

3.4 |x - b| <= a = -a < x - b < a

3.4.1 |x - b| >= a = b - a < x < a - b

3.5 |f(x)| = a <==>
f(x) = a or f(x) = -a

3.5.1 |f(x)| = |g(x)| <==>
(f(x))^2 = (g(x))^2

3.6 Mod graphs will never go below the x-axis