Cov Lus Nug Piv Txwv Sib Tham Txog Cov Haujlwm Rov Qab
Lub inverse function yog ib lub tswv yim tseem ceeb hauv kev lej, uas feem ntau pom ntawm ntau qib kev kawm. Lub tswv yim no pab peb nkag siab yuav ua li cas "invert" ib lub function, lossis nrhiav ib lub function uas tsim cov nqi pib ntawm cov zis ntawm lub function thawj. Hauv tsab xov xwm no, peb yuav tshawb nrhiav lub tswv yim ntawm inverse functions nrog ntau yam piv txwv teeb meem thiab cov txheej txheem daws teeb meem.
Kev Nkag Siab Yooj Yim Txog Cov Kev Ua Haujlwm Rov Qab
Lub luag haujlwm inverse, feem ntau yog cim los ntawm \( f^{-1} \), yog ib lub luag haujlwm uas rov qab tus nqi qub ntawm lub luag haujlwm \( f \). Hais yooj yim, yog tias \( f(x) = y \), ces \( f^{-1}(y) = x \).
Piv txwv li, xav tias koj muaj lub luag haujlwm \( f(x) = 2x + 3 \). Yog tias koj ntsaws rau hauv tus nqi \( x = 2 \), ces qhov tshwm sim yog \( f(2) = 2(2) + 3 = 7 \). Lub luag haujlwm rov qab ntawm \( f \), uas peb qhia los ntawm \( f^{-1}(x) \), yuav tsum rov qab peb mus rau tus nqi qub yog tias peb ntsaws rau hauv 7: \( f^{-1}(7) = 2 \).
Cov Kauj Ruam Nrhiav Kev Ua Haujlwm Rov Qab
Nov yog cov kauj ruam dav dav los nrhiav qhov kev ua haujlwm rov qab ntawm qhov kev ua haujlwm \( f(x) \):
1. Hloov \( f(x) \) nrog \( y \):
Piv txwv li, \( f(x) = 2x + 3 \), peb sau ua \( y = 2x + 3 \).
2. Hloov qhov chaw ntawm \( x \) thiab \( y \):
Yuav nrhiav qhov inverse, peb pauv \(x\) thiab \(y\) kom tau \(x = 2y + 3\).
3. Daws qhov sib npaug rau \( y \):
Peb daws qhov sib npaug \( x = 2y + 3 \) rau \( y \):
\[
\begin{aligned}
x = 2y + 3 \\
x – 3 &= 2y \\
y = x – 3/2
\end{aligned}
\]
4. Sau qhov Inverse Function:
Qhov kev ua haujlwm rov qab \( f^{-1}(x) \) ntawm \( f(x) = 2x + 3 \) yog \( f^{-1}(x) = \frac{x – 3}{2} \).
Tam sim no, cia peb nkag siab lub tswv yim yooj yim no nrog qee qhov teeb meem piv txwv.
Cov Lus Nug Piv Txwv thiab Kev Sib Tham
Piv txwv lus nug 1
Lus Nug: Nrhiav qhov kev ua haujlwm rov qab ntawm \( f(x) = \frac{1}{x – 4} \).
Kev Sib Tham:
1. Hloov \( f(x) \) nrog \( y \):
\[
y = \frac{1}{x – 4}
\]
2. Hloov qhov chaw ntawm \( x \) thiab \( y \):
\[
x = \frac{1}{y – 4}
\]
3. Daws qhov sib npaug rau \( y \):
\[
\begin{aligned}
x & = \frac{1}{y – 4} \\
xy = 1 \\
xy – 4x &= 1 \\
xy – 4x &= 1 \\
y – 4 &= \frac{1}{x} \\
y = \frac{1}{x} + 4
\end{aligned}
\]
4. Sau qhov Inverse Function:
Tus inverse function \( f^{-1}(x) \) yog \( f^{-1}(x) = \frac{1}{x} + 4 \).
Piv txwv lus nug 2
Lus Nug: Nrhiav qhov kev ua haujlwm rov qab ntawm \( g(x) = 3 - 5x \).
Kev Sib Tham:
1. Hloov \( g(x) \) nrog \( y \):
\[
y = 3 – 5x
\]
2. Hloov qhov chaw ntawm \( x \) thiab \( y \):
\[
x = 3 – 5y
\]
3. Daws qhov sib npaug rau \( y \):
\[
\begin{aligned}
x = 3 – 5y \\
x – 3 &= -5y \\
y = \frac{3 – x}{5}
\end{aligned}
\]
4. Sau qhov Inverse Function:
Tus inverse function \( g^{-1}(x) \) yog \( g^{-1}(x) = \frac{3 – x}{5} \).
Piv txwv lus nug 3
Lus Nug: Yog tias \( h(x) = \sqrt{x + 2} \), nrhiav qhov kev ua haujlwm rov qab \( h^{-1}(x) \).
Kev Sib Tham:
1. Hloov \( h(x) \) nrog \( y \):
\[
y = x + 2
\]
2. Hloov qhov chaw ntawm \( x \) thiab \( y \):
\[
x = y + 2
\]
3. Daws qhov sib npaug rau \( y \):
\[
\begin{aligned}
x & = y + 2} \\
x^2 &= y + 2 \\
y &= x^2 – 2
\end{aligned}
\]
4. Sau qhov Inverse Function:
Qhov kev ua haujlwm rov qab \( h^{-1}(x) \) yog \( h^{-1}(x) = x^2 – 2 \).
Piv txwv lus nug 4
Lus Nug: Nrhiav qhov kev ua haujlwm rov qab ntawm \( k(x) = \ln(x – 1) \) (nrog \( x > 1 \)).
Kev Sib Tham:
1. Hloov \( k(x) \) nrog \( y \):
\[
y = ln(x – 1)
\]
2. Hloov qhov chaw ntawm \( x \) thiab \( y \):
\[
x = ln(y – 1)
\]
3. Daws qhov sib npaug rau \( y \):
\[
\begin{aligned}
x &= \ln(y – 1) \\
e^x &= y – 1 \\
y &= e^x + 1
\end{aligned}
\]
4. Sau qhov Inverse Function:
Tus inverse function \( k^{-1}(x) \) yog \( k^{-1}(x) = e^x + 1 \).
Xaus
Kev nkag siab txog cov haujlwm rov qab yuav tsum tau xyaum thiab nkag siab ib kauj ruam zuj zus txog lub tswv yim thiab nws cov kev siv. Cov txheej txheem tseem ceeb suav nrog kev sib pauv cov hloov pauv, daws cov kab zauv, thiab sau qhov tshwm sim kawg ua ib qho haujlwm rov qab. Kev kawm txog ntau yam teeb meem piv txwv, xws li cov uas tau tham saum toj no, tuaj yeem pab peb txhim kho peb cov txuj ci hauv kev txheeb xyuas thiab nkag siab txog lub tswv yim ntawm cov haujlwm rov qab.
Los ntawm kev xyaum ua thiab kev nkag siab txog ntau yam teeb meem piv txwv, peb yuav muaj peev xwm daws ntau hom teeb meem uas cuam tshuam nrog cov haujlwm rov qab nrog kev ntseeg siab ntau dua.
