Verkettung zweier Funktionen

Die Verkettung zweier Funktionen y=f(x) und y=g(x) zu y = (fog)(x) = f(g(x))
wird räumlich veranschaulicht.
Wie müsste die vertauschte Verkettung  y= g(f(x)) aussehen?

x

v(x)

v(x)

u(v(x_0))

u(v(x))

u(v(x_0))

Verkettung von Funktionen

Bewege x_0

v(x) =

u(x) =

a) Probiere verschiedene Funktionstypen
b) Vertausche die beiden Funktionen

h

x_0+h

$f = u \circ v$

x + h

v(x+h)

u(v(x+h))

$s_{u \circ v}$

s_v

s_u

\cdot

=

\cdot

=

f(x)

f(x+h)

$f = u \circ v$

u

v

f = u \circ v

f_2(x) = ℯ^((0.2 * x))
x_0 = Point(xAxis)
f_1(x) = (1 / 2 * x^(2)) + 1
c = Line(x_0, yAxis)
A: (x(x_0), f_1(x(x_0)), 0)
d = Line(A, xAxis)
B = Intersect(d, yAxis)
e = Line(B, zAxis)
f = CurveCartesian(0, t, f_2(t), t, 0, 50)
C: f(y(B))
k = Line(C, yAxis)
D = Intersect(zAxis, k)
m = Plane((0, 0, 0), (1, 0, 0), (0, 0, 1))
n = Line(D, xAxis)
p = Line(x_0, zAxis)
E = Intersect(n, p)
g(x) = f_2(f_1(x))
Eingabefeld1 = Textfield(f_1)
Eingabefeld2 = Textfield(f_2)
v = Segment(A, B)
Text2: "v(x) = " + (LaTeX[f_1]) + ""
Text3: "u(x) = " + (LaTeX[f_2]) + ""
h0 = 1 (slider from 0 to 1, step 0.1)
H: x_0 + (h0, 0, 0)
x0: x(x_0)
mv: (f_1(x0 + h0) - f_1(x0)) / h0
mu: (f_2(f_1(x0 + h0)) - f_2(f_1(x0))) / (f_1(x0 + h0) - f_1(x0))
h = CurveCartesian(t, 0, g(t), t, 0, 50)
tmv: "m_v(x_0,x_0+h) = " + (LaTeX[mv]) + ""
mf: (g(x0 + h0) - g(x0)) / h0
T23: C + (0, 0, 2)
A_h: A + (h0, (mv * h0), 0)
s_v = Line(A, A_h)
X_h: x_0 + (h0, 0, 0)
B_h: (0, f_1(x0 + h0), 0)
C_h: B_h + (0, 0, f_2(f_1(x0 + h0)))
D_h: (0, 0, g(x0 + h0))
E_h: X_h + (0, 0, g(x0 + h0))
s_u = Line(C, C_h)
r = Segment(X_h, A_h)
s = Segment(A_h, B_h)
u = Segment(x_0, A)
w = Segment(B, C)
t = Segment(B_h, C_h)
i = Segment(C, D)
u_1 = Segment(C_h, D_h)
j = Segment(E, D)
w_1 = Segment(E_h, D_h)
v_1 = Segment(X_h, E_h)
l = Segment(x_0, E)
s_f = Line(E, E_h)
S_f = Point(s_f)
S_v = Point(s_v)
S_u = Point(s_u)
a_v: f_1'(x0)
a_u: f_2'(f_1(x0))
a_f: g'(x0)
Text4: "f(x) = (u∘v)(x) = u(v(x)) = " + (LaTeX[g]) + ""
A_a: A + (1, a_v, 0)
C_a: C + (0, 1, a_u)
E_a: E + (1, 0, a_f)
q = Line(A, A_a)
g_1 = Line(C, C_a)
h_1 = Line(E, E_a)
Text7: "m_v(x,x+h) \approx " + (LaTeX[mv]) + ""
Text9: "m_{u \circ v}(x,x+h) \approx " + (LaTeX[mf]) + ""
Text10: "v'(x) = " + (LaTeX[a_v]) + ""
Text11: "u'(v(x)) \approx " + (LaTeX[a_u]) + ""
Text12: "(u \circ v)'(x) \approx " + (LaTeX[a_f]) + ""
Text17: "Änderungsraten für x = " + x0 + ""
Text8: "m_u(v(x),v(x+h)) \approx " + (LaTeX[mu]) + ""
D': D
D_h': D_h
P_f: (4, 0, g(4))
P_u: (0, 6, f_2(6))
P_v: (3.5, f_1(3.5), 0)