# Lesson Functional Notation

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 4.09   Functional Notation            from Basic Algebra: One Step at a Time ฉ 2002-2011 P. 379-384 Dr. Robert J. Rapalje Seminole State College of Florida ANSWERS TO ALL EXERCISES ARE INCLUDED AT THE END OF THIS PAGE There is a special notation called functional notation that is frequently used in mathematics when one variable is described in terms of another.  The notation f(x) [read f of x] is often used to name a second variable.  Instead of writing y = 3x + 2 you may write f(x) = 3x + 2 or g(x) = 3x + 2 or perhaps even y(x) = 3x + 2.  Any letter may be used.  This notation indicates that f or  g or y  is a function of the variable x, which means that it can be expressed in terms of x.  To find the value of f(2), just replace each x with the value 2.  To find the value of f(4), replace each x in the given formula  with the value 4.  To find the value of f(-3), replace each x in the formula with the value -3.  Notice that f(x) does NOT mean to multiply f times x.   EXAMPLE 1.       Given f(x) = 2x + 3.  Find the values of  a) f(0), b) f(7), c) f(-5). Solution:            a)       f(0) means that x = 0.  Replace the x with the value of 0.     f(x) = 2x + 3     f(0) = 2(0) + 3            =   3        b)     f(7) means that x = 7.  Replace the x with the value of 7.     f(x) = 2x + 3     f(7) = 2(7) + 3            = 14 +  3  or 17           c)     f(-5) means that x = -5.  Replace the x with the value of -5.     f(x)  = 2x + 3     f(-5) = 2(-5) + 3             = -10 +  3  or -7   EXAMPLE 2.       Given g(x) = -6x2 + 3x - 5.  Find the values of  a) g(2), b) g(-5). Solution:            a)      g(2) means that x = 2.  Replace each x with the value of 2.                 g(x) =  -6x2  +  3x - 5                 g(2) = -6ื2 2 + 3ื2 - 5                        =  -6ื 4  +  6  -  5            =   -24   + 1  or  -23          b)      g(7) means that x = -5.                 g(x)   = -6x2 + 3x - 5                 g(-5) = -6ื(-5) 2 + 3ื(-5) - 5                         =  -6ื 25  -  15    -  5                         =   -150   -  20    or   -170   379   EXERCISES.        Complete the following.                         1.         f(x) = 3x + 2                                        2.         g(x)  = -3x + 5    a)      f(0) = 3( 0 ) + 2                                       a)      g(0)  = -3(       ) + 5                   = _________                                                     = ญญญญ__________    b)      f(2) =  3(  2  )  + 2                                   b)      g(2)  =  -3(      ) + 5                    =                                                                       =    c)      f(4) =  3(      )  +  2                                  c)      g(4)  =  -3(     ) + 5                    =                                                                       =    d)      f(-3) =                                                     d)      g(-3) =                     =                                                                       =    e)      f(\$) = 3(       ) + 2                                    e)      g(\$)   = __________    f)       f( * ) = __________                                f)      g( * ) = __________    g)      f(###) = __________                              g)     g(###) = __________    h)      f(Junk) = __________                            h)     g(Junk) = __________   3.         h(x) = -2x - 4                                       4.         y(x) = -4x + 6      a)      h(0) = -2(      ) - 4                                    a)      y(0) = -4(     ) + 6                     =                                                                      =     b)      h(2)  =                                                     b)      y(2) =                      =                                                                      =    c)      h(4) =                                                      c)      y(4)  =                     =                                                                       =    d)      h(-3) =                                                    d)      y(-3) =                     =                                                                       =    e)      h(Junk) =                                               e)      y(Junk)   = 380    5.       f(x) = x2 + 3x + 4                                 6.       g(x) = x2 + 2x - 4    a)      f(0)  = ( 0 )2 + 3( 0 )  + 4                        a)    g(0)  = (      )2 + 2(      ) - 4                     =                                                                     =             b)      f(4)  = ( 4 )2 + 3( 4 )  + 4                         b)    g(4) = (       )2 + 2(       )  - 4                     =                                                                     =                     =                                                                     =    c)      f(-5) =  (         )2 + 3(        )  + 4               c)   g(-5) = (       )2 + 2(       )  - 4                     =                                                                     =                     =                                                                     =    d)      f(\$)  =                                                      d)   g(\$)   =    e)      f(Junk) =                                                 e)  g(Junk) =     7.       f(x)      = -x2 + 4x - 3                            8.       g(x)      = -2x2 - 2x + 4    a)      f(0)      = - (      )2 + 4(      )  - 3               a)    g(0)      = -2(      )2 - 2(      ) + 4                         =                                                                      =                                 =                                                                      =    b)      f(2)       = - (       )2 + 4(       )  - 3           b)    g(2)       = -2(       )2 - 2(       )  + 4                         =                                                                      =                         =                                                                      =    c)      f(4)      = - (       )2 + 4(       )  - 3             c)    g(4)      =                         =                                                                      =                         =                                                                      =    d)      f(-2)     =                                                 d)     g(-2)     =                         =                                                                      =                         =                                                                      =    e)      f( # )    =                                                 e)      g(π)     =  381   9.        f(x) = 2x - 5                                         10.        g(x)    = -4x + 6      a)     f(\$) = 2(       ) - 5                                     a)      g(\$)    = ____________     b)   f(3x) = 2( 3x ) - 5                                      b)      g(3x)  = -4(        )  + 6         = _________                                                        = ____________     c)   f(5y) = 2(           ) - 5                                 c)       g(5y)  = -4(         )  +  6        = _________                                                        = ____________     d)   f(5y + 3) = 2( 5y+3 ) - 5                           d)   g(5y + 3)    = -4(             )  + 6              = _______________                                          = _________________              = _______________                                          = _________________     e)   f(5x - 4)  = ______________                     e)   g(5x - 4)   = _________________              = _______________                                          = _________________              = _______________                                          = _________________         11.       f(x) = 5x + 7                                        12.       g(x) = -4x - 5     a)   f(x + 2) =   5( x + 2 ) + 7                           a)   g(x + 2)  = -4(              )  - 5            =   _______________                                       = _________________            =   _______________                                       = _________________     b)   f(-3x + 7) = 5(               )  +  7                  b)   g(-3x + 7) = _________________            = _______________                                          = _________________            = _______________                                          = _________________     c)   f(x2 - 3) = _______________                   c)   g(x2 - 3)   =  _________________            = _______________                                          = _________________            = _______________                                          = _________________     d)   f(4x + 3y) = ______________              d)   g(4x + 3y)   = _________________                = _______________                                      = _________________                = _______________                                      = _________________     e)  f(7y2 + 3y - 4) = _________________   e)   g(7y2 + 3y - 4)  = _________________                     = _________________                                    = _________________                     = _________________                                    = _________________ 382 In the next example and exercises, notice that f(x) and g(x) are involved in the same problems.   EXAMPLE 2.             Given   f(x) = 5x - 2      and         g(x) = -3x + 4, find a) f [f(x)],    b) g [g(x)],   c) f [g(x)],    and   d) g [f(x)]. Solution. Remember?      f [ (Junk)]  = 5(          ) - 2                        g [ (Junk)] = - 3 (           ) + 4 a)   f [ f(x) ]  = 5(  f(x)     ) - 2                b)  g [ g(x) ]     =  - 3 (   g(x)   ) + 4                     = 5( 5x - 2  ) - 2                                         =  - 3 (  -3x + 4  ) + 4         = 25x - 10 - 2                                             =  9x - 12 + 4         = 25x  - 12                                                 =  9x - 8   c) f [ g(x) ]  = 5(  g(x)     ) - 2                d)  g [ f(x) ]      =  - 3 (   f(x)   ) + 4                    = 5( -3x + 4  ) - 2                                        =  - 3 (  2x - 3  ) + 4        = - 15x + 20 - 2                                           = - 6x + 9 + 4        = -15x  + 18                                                = - 6x + 13 EXERCISES.               13.                f(x) = 3x + 2                                                    g(x) = 2x - 3 a) f [ (Junk)] = 3(          ) + 2                            b)  g [ (Junk)] = 2 (           ) - 3 c)   f [ f(x) ]   = 3(  f(x)     ) + 2                         d)   g [ g(x) ]  =  2 (   g(x)   ) - 3           = 3(              ) + 2                                              =  2(            ) - 3           = ______________                                           = _____________           = ______________                                           = _____________ e)  f [ g(x) ]   = 3(  g(x)     ) + 2                         f)   g [ f(x) ]    =  2 (   f(x)   ) - 3          = 3(              ) + 2                                                =  2(            ) - 3          = ______________                                             = _____________          = ______________                                             = _____________ 383 14.                f(x) = - 5x + 2                                                  g(x) = -7x - 8 a) f [ (Junk)]     = ______________                b)  g [ (Junk)]   =   ___________   c) f [ f(x) ]        =                                               d)    g[ g(x) ]   =                 e) f [ g(x) ]  =                                                     f)   g [ f(x) ]   =                    15.       f(x) = 3x + 2                            g(x) = x2 - 5x + 3                    h(x) =  a)    f [g(x)] =                                                    b)     g [f(x)] =                                    c)     h [f(x)] =                                                  d)     h [g(x)] =         16.       f(x) =  3x - 2                            g(x) = x2 +  5x - 3                   h(x) =  a)     f [g(x)] =                                                  b)    g [f(x)] =                                     c)     h [f(x)] =                                                   d)    h [g(x)] =   384   ANSWERS 4.09  p. 379 - 384:  1a) 2, b) 8, c) 14, d) -7, e) 3\$+2, f) 3*+2, g) 3(###)+2,   h) 3(Junk)+2;   2a) 5, b) -1, c) -7, d) 14, e) -3\$+5, f) -3*+5, g) -3(###)+5, h) -3(Junk)+5;  3a) -4, b) -8, c) -12, d) 2, e) -2(Junk)-4;  4a) 6, b) -2, c) -10, d) 18, e) -4(Junk)+6;  5a) 4, b) 32, c) 14, d) \$2+3\$+4, e) (Junk)2+3(Junk)+4 ;  6a) -4, b) 20, c) 11, d) \$2+2\$-4, e) (Junk)2+2(Junk)-4;  7a) -3, b) 1, c) -3, d) -15, e) - #2+4# - 3;      8a) 4, b) -8, c) -36, d) 0, e) -2π2-2π+4;  9a) 2\$-5, b) 6x-5, c) 10y-5, d) 10y+1, e) 10x-13;  10a) -4\$+6, b) -12x+6, c) -20y+6, d) -20y-6, e) -20x+22;             11a) 5x+17, b) -15x+42, c) 5x2-8, d) 20x+15y+7, e) 35y2+15y-13; 12a) -4x-13, b) 12x-33, c) -4x2+7, d) -16x-12y-5, e) -28y2-12y+11;             13a) 3(Junk)+2, b) 2(Junk)-3, c) 9x+8, d) 4x-9, e) 6x-7, f) 6x+1;             14a) -5(Junk)+2, b) -7(Junk)-8, c) 25x-8, d) 49x+48, e) 35x+42, f) 35x-22; 15a) 3x2-15x+11, b) 9x2-3x-3, c), d) ; 16a) 3x2+15x-11, b) 9x2+3x-9, c), d) .    Return to main page       Math in Living C O L O R !!
 Dr. Robert J. Rapalje Altamonte Springs Campus Contact me at: Phone number:  NONE Retired!! OFFICE:          NONE Copyright ฉ Seminole State College of Florida, 1997

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