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Composite Functions
Functions of Functions
10/11/2013 Composite Functions 22
Fencing a Square Lot A square 2-acre yard is to be fenced How many feet of fencing is needed for each side? (1 acre = 43,560 sq.ft.) Solution:
Converting acres to square feet: f(x) = 43560x sq.ft.
f(2) = 43560(2) = 87120 sq.ft. = y sq.ft.
Composition of Functions
Let x = no. of acres y = no. of square ft.
10/11/2013 Composite Functions 33
Fencing a Square Lot Solution:
f(2) = 43560(2) = 87120 sq.ft. = y sq.ft. Converting sq.ft. to side length in feet:
g(y) = y1/2 ft.
g(f(2)) = g(43560(2))
= (43560(2))1/2
= (87120)1/2
= 295.16 ft.
Composition of Functions
10/11/2013 Composite Functions 4
Fencing a Square Lot Solution:
f(2) = y sq.ft.
g(f(2)) = 295.16 ft
4
y Square Feet
x Acres
Composition of Functions
● ● ●2 87120 295.16
f g
?
Feetg(f(x))
10/11/2013 Composite Functions 5
Functions of Functions If y = f(x) and y is in the domain of g, then g(y) = g(f(x)) is called a composite function of x
g is a function of a function
5
Composition of Functions
10/11/2013 Composite Functions 6
Functions of Functions Definition
For functions f and g, with g defined on the range of f , the function
is the composite function of g composed with f
6
Composition of Functions
(g f)(x) = g(f(x))°
Note: g f° is NOT the product of g and f
10/11/2013 Composite Functions 7
Functions of Functions Domain of
{ x | x is in domain f and f(x) is in domain g }
{ x | x dom f ; f(x) dom g }
Domain of
7
Composition of Functions
is a subset of domain f
(g f)(x) = g(f(x))° is
(g f)(x)°
OR
10/11/2013 Composite Functions 88
Stopping a Vehicle Find the reaction distance r(x) = tx traveled in estimated reaction time t = 2.5 secs at velocity x = 60 mph
Converting mph to ft/hr:
h(x) = 5280x ft/hr
h(60) = 5280(60)
= 316800 ft/hr
= y ft/hr
Composition of Functions
10/11/2013 Composite Functions 99
Stopping a Vehicle h(60) = 316800 ft/hr = y ft/hr Converting ft/hr to ft/sec: g(y) = y/3600 ft/sec
g(h(60)) = g(31600)
= 316800/3600
= 88 ft/sec
= z ft/sec
Composition of Functions
10/11/2013 Composite Functions 1010
Stopping a Vehicle h(60) = 316800 ft/hr = y ft/hr g(y) = g(h(60)) = 88 ft/sec = z ft/sec Converting ft/sec to feet:
f(z) = (5/2)z ft Thus reaction distance at 60 mph is
r(60) = f(g(h(60)))
= f(g(316800))
= f(88)
= (5/2)88 ft = 220 ft
Composition of Functions
10/11/2013 Composite Functions 1111
Stopping a Vehicle h(60) = 316800 ft/hr = y ft/hr g(y) = g(h(60)) = 88 ft/sec = z ft/sec f(z) = (5/2)z ft = 220 ft r(60) = f(g(h(60))) = 220 ft
Composition of Functions
z ft/sec y ft/hr x mi/hr
● ● ●60 316800 88
h g
r(x)
●f
220
feet
10/11/2013 Composite Functions 12
Composite Diagram:
12
Dom f
Composition of Functions
Range f
f g
Dom g
Range ga f(a)
bg(b)
Remember:
Question:Is Dom g = Range f ? … not necessarily ! b Dom g but b Range ff(x) ≠ b for any x but g(b) Range ga Dom f but f(a) Dom g
(version 1)g f°
Range is a subset of Range gg f °
Dom is a subset of Dom fg f °
So ( )(a) = g(f(a)) does not exist g f °
x f(x) g(f(x))● ● ●
°Dom g f °Range g f°g f
Where are Dom g f and Range g f ?° °
10/11/2013 Composite Functions 1313
Domain f
Composite Diagram
Composition of Functions
Range f
● ● ●x f(x) g(f(x))
f gRange ga f(a)
b g(b)
(version 2)g f°
°Dom g f Range g f°
Domain g
Range is a subset of Range gg f °
Dom is a subset of Dom fg f °
° g f
10/11/2013 Composite Functions 14
Example:
14
Composition of Functions
f(x) = x + 2 g(x) = 5x + x
= g(x + 2)= 5(x + 2) + x + 2
Dom f = R Dom g = { x x ≥ 0 }Range f = R Range g = { x x ≥ 0 }
(g f)(x) = g(f(x))°
10/11/2013 Composite Functions 15
Example:
15
Composition of Functions
f(x) = x + 2 g(x) = 5x + x
= [ –2 , )= [ 0 , )
= 5(x + 2) + x + 2 (g f)(x) °
Dom g f = { x x ≥ –2 } °Range g f = { x x ≥ 0} °
10/11/2013 Composite Functions 16
Example:
16
Composition of Functions
f(x) = x + 2 g(x) = 5x + x
= f ( 5x + x )= 5x + x + 2
(f g)(x) = f(g(x))°
Question:
(g f)(x)° = 5(x + 2) + x + 2
°How do g f and differ ?f g°
10/11/2013 Composite Functions 17
Example:
17
Composition of Functions
f(x) = x + 2 g(x) = 5x + x
= 5x + x + 2
(f g)(x) °
= [ 2 , )Range f g
°
(g f)(x)° = 5(x + 2) + x + 2
Dom f g ° = [ 0 , )
= [ –2 , )Dom g f
° = [ 0 , )Range g f °
What if f(x) = x2 – 4x – 2 = x + 2 , for x ≠ 2 ?
10/11/2013 Composite Functions 18
Example:
18
Composition of Functions
f(x) = x + 2 g(x) = 5x + x
f(x) = x2 – 4x – 2 = x + 2 , for x ≠ 2 ?
Dom f g ° = { x | x ≥ 0 , x ≠ .7350 }
= { x | x ≥ –2 , x ≠ 2 }Dom g f
°= { x | x ≥ 0 , x ≠ 22 }Range g f °
Range f g
° = { x | x ≥ 2, x ≠ 13.414 }
10/11/2013 Composite Functions 1919
Domain f
Example:
Composite Diagram
Composition of Functions
g(x) = 5x + x= g(x + 2) = 5(x + 2) + x + 2
Range f
● ● ●x f(x) g(f(x))
f g
Domain g
Range g–5 –3
4 22
f(x) = x2 – 4x – 2 = x + 2 , for x ≠ 2, f(x) ≠ 4
f
(g f)(x) = g(f(x))°
g f°
Dom g f = { x x ≥ –2 , x ≠ 2 }
° Range g f ={ x x ≥ 0 , x ≠ 22 }
°
10/11/2013 Composite Functions 2020
Composition Versus Multiplication NOTE: Example : Let f(x) = x2 and g(x) = x1/2
Composition of Functions
= g(x2)= (x2)1/2 = |x|
... then
(gf)(x) = g(x) • f(x) = x1/2 • x2 = x5/2 ... which is NOT |x| !!
, for all x
(g f)(x) ≠ (gf)(x)°
But …
(g f)(x) = g(f(x)) °
But …
10/11/2013 Composite Functions 2121
Composition Versus Multiplication Is it possible that
Composition of Functions
... well ... = f(g(x)) = f(x1/2)= (x1/2)2
Question:
Only if their domains are equal ! Are they ?
, for x ≥ 0
?(g f)(x) = (f g)(x)
° °
Are and the same function ?
g f ° f g °
= (g f)(x)
°
(f g)(x)
°
= x
10/11/2013 Composite Functions 22
Find
22
Combining Functions
Note:Output of the first function is input to the second
= f(1) = 3 = -2= g(-1)
and
graphically (g f)(–3) °(f g)(2)
°
= g(f(-3))(g f)(-3)
°= f(g(2))(f g)(2) °
f(1) = 3
-3-1
-2
3
1
1 2 3x
y
●●
●
● g(2) = 1
y = f(x) y =g(x)
-3-1
-2
3
1
1 2 3x
y
y = f(x) y =g(x)
●
●
●●f(-3) = -1
10/11/2013 Composite Functions 2323
Combining Functions
Find
= f(g(2))
= g(f(-3))
Note: Output of the first function is input to the second
= f(1) = 3
= -2= g(-1)
and in tabular form
(g f)(–3) °(f g)(2) °
x –3 –2 –1 0 1 2 3
f(x) –1 0 1 2 3 4 5 g(x) 5 1 –2 –3 –2 1 5 –2
–1
–1 3
1
–3 2 1
°(f g)(2)
°(g f)(-3)
10/11/2013 Composite Functions 2424
Think about it !