Mehlala ea Lipotso tse Buisanang ka Mesebetsi e Fetohileng
Mosebetsi o fapaneng ke mohopolo oa motheo lipalo, o atisang ho fumanoa maemong a fapaneng a thuto. Khopolo ena e re thusa ho utloisisa mokhoa oa ho "fetola" mosebetsi, kapa ho fumana mosebetsi o hlahisang boleng ba pele ba tlhahiso ea mosebetsi oa pele. Sehloohong sena, re tla hlahloba ka botlalo mohopolo oa mesebetsi e fapaneng ka mathata a fapaneng a mehlala le mekhoa ea tharollo.
Kutloisiso ea Motheo ea Mesebetsi e Fetohileng
Mosebetsi o fapaneng, o atisang ho bontshwa ke \( f^{-1} \), ke mosebetsi o kgutlisang boleng ba pele ba mosebetsi \( f \). Ka bokhutšoanyane, haeba \( f(x) = y \), ebe \( f^{-1}(y) = x \).
Mohlala, a re re o na le mosebetsi \( f(x) = 2x + 3 \). Haeba o kenya boleng \( x = 2 \), sephetho ke \( f(2) = 2(2) + 3 = 7 \). Mosebetsi o fapaneng wa \( f \), oo re o bolelang ka \( f^{-1}(x) \), o lokela ho re kgutlisetsa boleng ba pele haeba re kenya 7: \( f^{-1}(7) = 2 \).
Mehato ea ho Fumana Mosebetsi o Fapaneng
Mehato e akaretsang ea ho fumana mosebetsi o fapaneng oa mosebetsi \( f(x) \) ke ena:
1. Nka sebaka sa \( f(x) \) ka \( y \):
Mohlala, \( f(x) = 2x + 3 \), re ngola jwalo ka \( y = 2x + 3 \).
2. Fetola maemo a \( x \) le \( y \):
Ho fumana se fapaneng, re fapanyetsana \( x \) le \( y \) ho fumana \( x = 2y + 3 \).
3. Rarolla equation bakeng sa \( y \):
Re rarolla equation \( x = 2y + 3 \) bakeng sa \( y \):
\[
\begin{align}
x &= 2y + 3 \\
x – 3 &= 2y \\
y &= \frac{x – 3}{2}
\end{align}
\]
4. Ngola Mosebetsi o Fetohileng:
Mosebetsi o fapaneng \( f^{-1}(x) \) wa \( f(x) = 2x + 3 \) ke \( f^{-1}(x) = \frac{x – 3}{2} \).
Jwale, ha re utlwisiseng mohopolo ona wa motheo ka mehlala e meng ya mathata.
Lipotso tsa Mehlala le Puisano
Mohlala oa Potso ea 1
Potso: Fumana mosebetsi o fapaneng wa \( f(x) = \frac{1}{x – 4} \).
Puisano:
1. Nka sebaka sa \( f(x) \) ka \( y \):
\[
y = \frac{1}{x – 4}
\]
2. Fetola maemo a \( x \) le \( y \):
\[
x = \frac{1}{y – 4}
\]
3. Rarolla equation bakeng sa \( y \):
\[
\begin{align}
x &= \frac{1}{y – 4} \\
xy &= 1 \\
xy – 4x &= 1 \\
xy – 4x &= 1 \\
y – 4 &= \frac{1}{x} \\
y &= \frac{1}{x} + 4
\end{align}
\]
4. Ngola Mosebetsi o Fetohileng:
Mosebetsi o fapaneng \( f^{-1}(x) \) ke \( f^{-1}(x) = \frac{1}{x} + 4 \).
Mohlala oa Potso ea 2
Potso: Fumana mosebetsi o fapaneng wa \( g(x) = 3 – 5x \).
Puisano:
1. Nka sebaka sa \( g(x) \) ka \( y \):
\[
y = 3 – 5x
\]
2. Fetola maemo a \( x \) le \( y \):
\[
x = 3 – 5y
\]
3. Rarolla equation bakeng sa \( y \):
\[
\begin{align}
x &= 3 – 5y \\
x – 3 &= -5y \\
y &= \frac{3 – x}{5}
\end{align}
\]
4. Ngola Mosebetsi o Fetohileng:
Mosebetsi o fapaneng \( g^{-1}(x) \) ke \( g^{-1}(x) = \frac{3 – x}{5} \).
Mohlala oa Potso ea 3
Potso: Haeba \( h(x) = \sqrt{x + 2} \), fumana mosebetsi o fapaneng \( h^{-1}(x) \).
Puisano:
1. Nka sebaka sa \( h(x) \) ka \( y \):
\[
y = \sqrt{x + 2}
\]
2. Fetola maemo a \( x \) le \( y \):
\[
x = \sqrt{y + 2}
\]
3. Rarolla equation bakeng sa \( y \):
\[
\begin{align}
x &= \sqrt{y + 2} \\
x^2 &= y + 2 \\
y &= x^2 – 2
\end{align}
\]
4. Ngola Mosebetsi o Fetohileng:
Mosebetsi o fapaneng \( h^{-1}(x) \) ke \( h^{-1}(x) = x^2 – 2 \).
Mohlala oa Potso ea 4
Potso: Fumana mosebetsi o fapaneng wa \( k(x) = \ln(x – 1) \) (ka \( x > 1 \)).
Puisano:
1. Nka sebaka sa \( k(x) \) ka \( y \):
\[
y = \ln(x – 1)
\]
2. Fetola maemo a \( x \) le \( y \):
\[
x = \ln(y – 1)
\]
3. Rarolla equation bakeng sa \( y \):
\[
\begin{align}
x &= \ln(y – 1) \\
e^x &= y – 1 \\
y &= e^x + 1
\end{align}
\]
4. Ngola Mosebetsi o Fetohileng:
Mosebetsi o fapaneng \( k^{-1}(x) \) ke \( k^{-1}(x) = e^x + 1 \).
Qetello
Ho utloisisa mesebetsi e fapaneng ho hloka ho itloaetsa le kutloisiso ea mohato ka mohato ea mohopolo le ts'ebeliso ea oona. Ts'ebetso e kholo e kenyelletsa ho fapanyetsana li-variable, ho rarolla li-equation, le ho ngola sephetho sa ho qetela e le mosebetsi o fapaneng. Ho ithuta mathata a fapaneng a mehlala, joalo ka a tšohliloeng kaholimo, ho ka re thusa ho ntlafatsa tsebo ea rona ea ho khetholla le ho utloisisa mohopolo oa mesebetsi e fapaneng.
Ka ho itloaetsa le kutloisiso e felletseng ea mathata a fapaneng a mehlala, re tla khona ho rarolla mefuta e fapaneng ea mathata a amanang le mesebetsi e fapaneng ka kholiseho e eketsehileng.
