Algebra and Function

Algebra and Function

1. Transformations

   • y = f(x) + a
     • Translation ( 0 , a )
       • Moves the graph up a units
   • y = f(x - a)
     • Translation ( a , 0 )
       • Subtracting a from x shifts the graph to the right
   • y = -f(x) is a reflection in the x-axis
     • y = f(-x) is a reflection in the y-axis
   • y = af(x) is a stretch in the y direction by a
     • y = f(ax) is a stretch in the x direction be a^-1
   • For y = f(|x|) for x > 0 and reflects in y to the right
     • For y = |f(x)| for y < 0 reflected in the line of the dotted x-axis

2. Functions

   • A function is defined by:
     • A rule connecting the range and domain sets
     • For each member of the domain, there is only one range value
   • A Function y = f(x)
     • One to One: One X value maps to one Y value
     • Many to One: More than one value of X maps to one value of Y
   • Composite Funtion
     • fg(x) = f(g(x)) Put g into f
       • The output of g becomes the input of f
     • Can only be formed in the example of fg(x). When the range of g is in the domain of f
   • Inverse Function
     • f^-1(x)
       • These can only exist when f(x) is a one to one mapping
     • The range of f is the domain of f^-1 and vice versa
       • The graph y=f(x) is the reflection of y=f^-1(x)
     • 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

   • |x| = x if x > 0
     • |x| = -x if x < 0
   • |x| < a = -a < x < a
     • |x| > a = x < -a or x > a
   • |x -a | = x - a for x >= a
     • |x - a| = -(x - a) = a - x for x
   • |x - b| <= a = -a < x - b < a
     • |x - b| >= a = b - a < x < a - b
   • |f(x)| = a <==> f(x) = a or f(x) = -a
     • |f(x)| = |g(x)| <==> (f(x))^2 = (g(x))^2
   • Mod graphs will never go below the x-axis

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