Algebra and Function

bubblesthelabrad
Mind Map by , created almost 5 years ago

A-Level Mathematics (Core 3) Mind Map on Algebra and Function, created by bubblesthelabrad on 02/10/2015.

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bubblesthelabrad
Created by bubblesthelabrad almost 5 years ago
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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