function theory
DESCRIPTION
Do you want to know how solving a function problem in mathematics? check this nice presentation out! Best recommended for senior high school student!!!!TRANSCRIPT
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BY: M. PRAHAS TOMI M. S.
XI SC-7SMAN 1 BOGOR
2009
M. PRAHASTOMI M. S.
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BASIC EXPLANATION SOME CASES ‘THIS FOR YOU’ SECTION
M. PRAHASTOMI M. S.
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FUNCTION COMPOSITION INVERS OF FUNCTION INVERS ALGORITHM INVERS OF FUNCTION COMPOSITION
M. PRAHASTOMI M. S.
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It Means the function of certain function(s) which given.
Simply we write the function composition above:
= (f o g o h o …) (x) = f [g {h (… (x)}]
M. PRAHASTOMI M. S.
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f(x) = ax + bg(x) = dx + ch(x) = mx + n
Case 1 (f o g) (x) = f [g(x)] = a [g(x)] + b = a (dx + c) + b
Case 2 (g o f) (x) = g [f(x)] = d [f(x)] + c = d (ax + b) + c
Case 3( f o g o h) (x) = f[ g{h(x)}] = f[g{mx + n}] = f[ d(mx + n) + c ] = a {d(mx + n) +c} + b
M. PRAHASTOMI M. S.
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Given: f(ax + b) = mx + ng(qx +r) = wx + z
(f o g) (x) = f[g(x)]
Firstly, we’ve ta find f (x). f(x) ≠ f(ax + b)
so, f [{a.(x-b)/a }+ b] = {m (x-b)/a} + n why (x-b)/a replacing x which suppose to be there?
Ex, y = ax + b
x= (y-b)/a
We change y become x, and replace it into the equation:f(x) = m { (x-b)/a } + n
M. PRAHASTOMI M. S.
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The same way with finding g (x)
g(x) = w{(x-r)/q} + n
so, (f o g) (x) = f[ g(x) ] = m [ {g(x) –b}/a ] + n
Algebraic count gives= f [g(x)] = (m/a)[{w(x-r) + q (b +z)}/q] + n
M. PRAHASTOMI M. S.
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f(ax+b)= [cx +d]c+d
g(wx+y)= [px + q]p+q
h(mx+n)= [rx + s]r+s
(g o h o f) (x) =…
M. PRAHASTOMI M. S.
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f(x) = ax + b
(f o g) (x) = cx + d
Find g(x) !
f[ g(x) ] = cx + d
f[ g(x) ] = cx+ d = a{g(x)} + b
g(x) = (cx + d –b)/a
M. PRAHASTOMI M. S.
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f(x) = ax + b
(g o f) (x) = cx + d
g(x) =...
g[ f(x) ] = cx +d
g[ ax + b ] = cx + d
We can find g(x) with 4th case way
g(x) = c [(x-b)/a] + d
M. PRAHASTOMI M. S.
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Symbolized: f -1 (x)
(f o f -1 ) (x) = ( f -1 o f ) (x)= x
M. PRAHASTOMI M. S.
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f(x) = ax + b
f -1 (x) = …
(f o f -1 ) (x) = f [f -1 (x) ] = a[f -1 (x) ] + b = x
f -1 (x) = (x-b)/a
M. PRAHASTOMI M. S.
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f(ax + b) = cx + d
f -1 (x) = …
f(x) = c{(x-b)/a} + d
(f o f -1 ) (x) = f [f -1 (x) ] = c{( f -1 (x) -b)/a} + d =x
f -1 (x) = a{(x-d)/c} + b
M. PRAHASTOMI M. S.
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f(x) = [ (ax + b)/(cx + d) ]t
f -1 (x) = …
(f o f -1 ) (x) = f [f -1 (x) ] = [ {a f -1 (x) + b}/{c f -1 (x) + d} ]t = x
x 1/t = {a f -1 (x) + b}/{c f -1 (x) + d}
cx1/t f -1 (x) + d x 1/t = a f -1 (x) + b
f -1 (x) = (-d x 1/t + b)/ (c x 1/t –a)
M. PRAHASTOMI M. S.
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f(x) = xLog (ax)/d f -1 (x) = …
(f o f -1 ) (x) = f [f -1 (x) ] = xLog [af -1 (x) ] / d = x
x x = [af -1 (x) ] / d
f -1 (x) = (dxx)/a
M. PRAHASTOMI M. S.
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f(x) = m Log [(ax-b)/( b -mn – c h )]ax
f -1 (mx) = …
M. PRAHASTOMI M. S.
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This is how computer know and count any function(s).Given case:f(x) = kx + p function algorithm 1 2x → kx → kx + pinverse algoritm 2 1x → x – p → (x-p)/k
Find the pattern of the following number!
M. PRAHASTOMI M. S.
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f(x) = (dxn + wz)pq
inverse algoritm?
x → x 1/pq → x1/pq – wz → (x 1/pq – wz)d → [(x1/pq – wz)d] 1/n
M. PRAHASTOMI M. S.
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f(x) = [rt(dxn + wz)pq] m
inverse algoritm?
x → x 1/m → (x 1/m )/rt → {(x 1/m )/rt}1/pq → {(x 1/m )/rt}1/pq – wz → [{(x 1/m )/rt}1/pq – wz]/d → [[{(x 1/m )/rt}1/pq – wz]/d] 1/n
M. PRAHASTOMI M. S.
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Figure out inverse of logarithm!
f -1 (2x-c) = 3p[rt +k(dxhn + wz)pq] m-p+2t
M. PRAHASTOMI M. S.
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for any given equation:
(f o g o h o …) -1 (x) = (... o h -1 o g -1 o f -1 ) (x)
M. PRAHASTOMI M. S.
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f(x) = ax + d
g(x) = mxn + p2/n
(f o g) -1 (x) = ...
g -1 (x) = [(x – p2/n)m] 1/n
f -1 (x) = (x –d)/a
(f o g) -1 (x) = (g -1 o f -1 ) (x) = g -1 [f -1 (x)]
= [(f -1 (x) – p2/n)m] 1/n
= [{(x –d)/a } - p2/n }m] 1/n
M. PRAHASTOMI M. S.
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f -1 (cx+d) = n(abx + mh+t ) m/n
g(x) = 3rx Log (ax)/d
Figure out (f -1 o g) -1 (x) !
M. PRAHASTOMI M. S.
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See you!
M. PRAHASTOMI M. S.