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) 
.
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