Combinations of Functions

Combining Functions in different ways:

Let f(x) = 4x - x2 and g(x) = 2x
    (a)  ( f + g )(x) = f(x) + g(x) = 4x - x2 + 2x  =  6x - x2
    (b)  ( f - g )(x) = f(x) + g(x) = 4x - x2 - 2x  =  2x - x2
    (c)  ( f · g )(x) =  f(x) g(x) = (4x - x2)(2x)  =  8x2 - 2x3
    (d)  ( f / g )(x) = f(x) / g(x) =  (4x - x2) / (2x) = 2 - (1/2)x
    (e)  f ( g (3)) or (f o g)(3)
                  g(3) = 2(3) = 6
                  f ( g (3)) = f(6) = 4(6) - 62 = 24 - 36 = -12
    (f)  f ( g (x)) = ( g o f )(x) = f(2x) = 4(2x) - (2x)2 = 8x - 4x2
 

Building composite functions:

Let f (x) = x3, g (x) = 2x - 1, and h (x) = |x|
     How is |(2x - 1)3| built?
     Answer:    h(f(g(x)))

     How is 2|x|3 -1 built?
     Answer:  g(f(h(x)))

     How is |2x3-1| built?
     Answer:  h(g(f(x)))
 

Domain:

Let  and .  Find the domain for f ( g (x)) and again for g ( f (x)).

        Domain for f(x): x ¹ -2
        Domain for g(x): 4 - x2 ³ 0
                                 (2 - x)(2 + x) ³ 0
                                                             -              +                  -
                                   Squiggly Line:  -------|----------------|------
                                                                   -2                      2
                         Answer:    -2 £ x £ 2

 f ( g (x)):
                                  
       x--------------------- Ö(4 - x2) -------------------- 1/(Ö(4 - x2) + 2)
                                    Ö(4 - x2) ¹ -2
                                       It never is; square roots are positive values.

                                       Domain:  -2 £ x £ 2  because these are the only values you can put into the first function.

  g ( f (x):
                                   
        x--------------------- 1/(x + 2)  -------------------- Ö(4 - [1/(x + 2)] 2)
                                     -2 £ 1/(x + 2) £ 2
                                1/(x + 2) ³ -2   and       1/(x + 2) £ 2
                          1/(x + 2) + 2 ³ 0            1/(x + 2) - 2 £ 0
                         1 + 2(x + 2) ³ 0             1 - 2(x + 2) £ 0
                               x + 2                              x + 2
                           1 + 2x + 4 ³ 0                 1 - 2x - 4 £ 0
                                 x + 2                               x + 2
                               2x + 5 ³ 0                       - 2x - 3 £ 0
                                 x + 2                                 x + 2

                                +     -       +                      -      +      -
Squiggly Line:          ---|------|-----                ----|-----|-----
                                 -5/2    -2                           -2    -3/2

                                 x £ -5/2, x > -2    and    -2 < x £ -3/2
                                        Domain:  -2 < x £ -3/2
 
 


Problems(No questions here yet)

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