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4.09 Functional Notation from Basic Algebra: One Step at a Time ฉ 2002 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
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) =
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) = 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) = _________________ = _________________ = _________________ = _________________ = _________________ 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 = ______________ = _____________ = ______________ = _____________
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)] =
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)
16a) 3x2+15x-11, b)
9x2+3x-9, c)
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