combining functions algebraically, composite functions, and decomposing functions!
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Combining functions algebraically, composite functions, and decomposing functions!. Onward to Section 1.4a…. Definition: Sum, Difference, Product, and Quotient of Functions. Let f and g be two functions with intersecting domains. Then for - PowerPoint PPT PresentationTRANSCRIPT
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Combining functions Combining functions algebraically, composite algebraically, composite
functions, and functions, and decomposing functions! decomposing functions!
Onward to Section Onward to Section 1.4a…1.4a…
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Definition: Sum, Difference, Product, and Quotient ofFunctions
Let f and g be two functions with intersecting domains. Then forall values of x in the intersection, the algebraic combinations off and g are defined by the following rules:
Sum: f g x f x g x
In each case, the domain of the new function consists of allnumbers that belong to both the domain of f and the domain of g.As noted, the zeros of the denominator are excluded from thedomain of the quotient.
Difference: f g x f x g x Product: fg x f x g x
Quotient: ,
f xfx
g g x
0g x provided
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Guided PracticeGuided PracticeFor the given functions, find f + g, f – g, fg, f/g, and gg. Givethe domain of each.
2f x x 1g x x
First, find the domain of the original functions, and determinewhere these two domains intersect (overlap).
Domain of f: , Domain of g: 1,
Domain intersection: 1,
This intersection becomes the domain of all of the algebraiccombination functions!!!
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Guided PracticeGuided PracticeFor the given functions, find f + g, f – g, fg, f/g, and gg. Givethe domain of each.
with D:
f g x f x g x 2 1x x 1,
f g x f x g x 2 1x x
fg x f x g x 2 1x x
2f x x 1g x x
with D: 1,
with D: 1,
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Guided PracticeGuided PracticeFor the given functions, find f + g, f – g, fg, f/g, and gg. Givethe domain of each.
with D: f xf
xg g x
2
1
x
x
1,
with D: gg x g x g x 21x 1,
Can we simplify this last one???Can we simplify this last one???
2f x x 1g x x
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Guided PracticeGuided PracticeFor the given functions, find formulas for the functions f + g,f – g, and fg. Give the domain of all functions.
Domain of all 3 combination functions:
5f x x 3g x x
D : 5, D : ,
5 3f g x x x
5 3f g x x x
3 5fg x x x
5,
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Guided PracticeGuided PracticeFor the given functions, find formulas for f/g and g/f. Give thedomain of all functions.
2f x x 4g x x
2
4
f xx
g x
D : 2, D : 4,
2
4
x
x
D : 2,
4
2
g xx
f x
4
2
x
x
D : 2,
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Now on to Now on to composite composite functions?!functions?!
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Definition:Definition:Composition of FunctionsComposition of Functions
Let f and g be two functions such that the domain of f intersectsthe range of g. The composition of f and g, denoted f g, isdefined by the rule
The domain of f g consists of all x-values in the domain of gthat map to g(x)-values in the domain of f.
NOTE: In most cases, f g and g f are different functions!!!
f g x f g x
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A Few Practice Problems…A Few Practice Problems…
xf x e
For the given functions, find (f g)(x) and (g f)(x) and verify(both algebraically and graphically) that the two compositefunctions are not the same.
g x x
f g x f g x f x xe g f x g f x xg e xe
Now, how do we verify???Now, how do we verify???
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A Few Practice Problems…A Few Practice Problems…
2 1f x x
For the given functions, find (f g)(x) and (g f)(x) and give thedomain of each composition function.
g x x
f g x 2 1x
g f x 2 1x
0,D:
, 1 1, D:
Let’s check these with the calculator…Let’s check these with the calculator…
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A Few Practice Problems…A Few Practice Problems…
For the given functions, find (f g)(3) and (g f)(–2).
2 1f x x 2 3g x x
3 3f g f g
8
2 2g f g f
3
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Decomposing Functions…working Decomposing Functions…working backwards or undoing a composition…backwards or undoing a composition…
21 3 1 4h x x x
For each function h, find functions f and g such that h(x) = f(g(x)).
2 3 4f x x x 1g x x
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Decomposing FunctionsDecomposing Functions
3 1h x x
For each function h, find functions f and g such that h(x) = f(g(x)).
f x x 3 1g x x
Any other ways to solve this one?!?!Any other ways to solve this one?!?!
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Let’s do some Let’s do some modeling…modeling…
In the medical procedure known as angioplasty, doctors insert acatheter into a heart vein and inflate a small, spherical balloonon the tip of the catheter. Suppose the balloon is inflated at aconstant rate of 44 cubic millimeters per second.
In math-land, not fashion-land…In math-land, not fashion-land…
1. Find the volume after t seconds.V = 44V = 44tt
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Let’s do some Let’s do some modeling…modeling…
34π3r V
In the medical procedure known as angioplasty, doctors insert acatheter into a heart vein and inflate a small, spherical balloonon the tip of the catheter. Suppose the balloon is inflated at aconstant rate of 44 cubic millimeters per second.
In math-land, not fashion-land…In math-land, not fashion-land…
2. When the volume is V, what is the radius r ?
3 3
4π
Vr
33
4π
Vr
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Let’s do some Let’s do some modeling…modeling…
3 33 44 33
4π π
t tr
In the medical procedure known as angioplasty, doctors insert acatheter into a heart vein and inflate a small, spherical balloonon the tip of the catheter. Suppose the balloon is inflated at aconstant rate of 44 cubic millimeters per second.
In math-land, not fashion-land…In math-land, not fashion-land…
3. Write an equation that gives the radius r as a function of the time. What is the radius after 5 seconds?
44V t 33
4π
Vr and
At 5 seconds,At 5 seconds,r = 3.745 mmr = 3.745 mm
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Whiteboard PracticeWhiteboard PracticeFor the given functions, find formulas for the functions f + g,f – g, and fg. Give the domain of all functions.
21f x x 3g x x
f g x 2 3 4x x
f g x 2 2x x
( )fg x 3 25 7 3x x x
Domain of all five functions: ,
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Find f(g(x)) and g(f(x)). State the domain of each.
Whiteboard Whiteboard PracticePractice
2( ) 1f x x 1
( )1
g xx
2
1( ( )) 1
( 1)
: ( ,1) (1, )
f g xx
D
2
1( ( ))
2
: ( , 2) ( 2, 2) ( 2, )
g f xx
D
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Find f(x) and g(x) so that the function can be described as y=f(g(x)).
one possible solution…Homework: p. 127-128 1-23 odd
Whiteboard Whiteboard PracticePractice
2
3
( ) ( 1)
( )
f x x
g x x
3 2( 1)y x