Ngā Tauira Pātai e Matapaki ana i ngā Pānga Whakamuri
He ariā taketake te mahi whakahurihuri i roto i te pāngarau, e kitea pinepine ana i ngā taumata mātauranga rerekē. Mā tēnei ariā tātou e āwhina ki te mārama me pēhea te "hurihuri" i tētahi mahi, te kimi rānei i tētahi mahi e whakaputa ana i te uara tīmatanga o te putanga o te mahi taketake. I roto i tēnei tuhinga, ka tūhuratia e tātou te ariā o ngā mahi whakahurihuri me ngā tauira rapanga me ngā tikanga whakaoti rapanga.
Te Māramatanga Taketake mō ngā Mahi Whakamuri
Ko te mahi whakamuri, e tohuhia ana e \( f^{-1} \), he mahi e whakahoki ana i te uara taketake o tētahi mahi \( f \). He māmā noa iho, mēnā ko \( f(x) = y \), ko \( f^{-1}(y) = x \).
Hei tauira, me kī kei a koe te mahi \( f(x) = 2x + 3 \). Ki te mono koe i te uara \( x = 2 \), ko te hua ko \( f(2) = 2(2) + 3 = 7 \). Ko te mahi whakamuri o \( f \), e tohuhia ana e tātou mā te \( f^{-1}(x) \), me whakahoki mai i a tātou ki te uara taketake mēnā ka monohia e tātou te 7: \( f^{-1}(7) = 2 \).
Ngā Hipanga hei Kimi i te Mahi Whakamuri
Anei ngā mahi whānui hei kimi i te mahi whakamuri o tētahi mahi \( f(x) \):
1. Whakakapia te \( f(x) \) ki te \( y \):
Hei tauira, \( f(x) = 2x + 3 \), ka tuhia e mātou ko \( y = 2x + 3 \).
2. Whakawhitihia ngā tūranga o \( x \) me \( y \):
Hei kimi i te whakahuri, ka whakawhitihia e tātou a \( x \) me \( y \) kia whiwhi ai i a \( x = 2y + 3 \).
3. Whakaotia te whārite mō \( y \):
Ka whakaotihia e mātou te whārite \( x = 2y + 3 \) mō \( y \):
\[
\begin{align}
x &= 2y + 3
x – 3 &= 2y
y &= \frac{x – 3}{2}
\end{whakahāngai}
\]
4. Tuhia te Mahi Whakamuri:
Ko te mahi whakahuri \( f^{-1}(x) \) o \( f(x) = 2x + 3 \) ko \( f^{-1}(x) = \frac{x – 3}{2} \).
Nā, kia mārama tātou ki tēnei ariā taketake me ētahi tauira rapanga.
Ngā Pātai Tauira me te Kōrero
Tauira Pātai 1
Pātai: Kimihia te mahi whakamuri o \( f(x) = \frac{1}{x – 4} \).
Kōrero:
1. Whakakapia te \( f(x) \) ki te \( y \):
\[
y = \frac{1}{x – 4}
\]
2. Whakawhitihia ngā tūranga o \( x \) me \( y \):
\[
x = \frac{1}{y – 4}
\]
3. Whakaotia te whārite mō \( y \):
\[
\begin{align}
x &= \frac{1}{y – 4} \\
xy &= 1 \\
xy – 4x &= 1 \\
xy – 4x &= 1 \\
y – 4 &= \frac{1}{x} \\
y &= \frac{1}{x} + 4
\end{whakahāngai}
\]
4. Tuhia te Mahi Whakamuri:
Ko te mahi whakamuri \( f^{-1}(x) \) ko \( f^{-1}(x) = \frac{1}{x} + 4 \).
Tauira Pātai 2
Pātai: Kimihia te pānga whakamuri o \( g(x) = 3 – 5x \).
Kōrero:
1. Whakakapia te \( g(x) \) ki te \( y \):
\[
y = 3 – 5x
\]
2. Whakawhitihia ngā tūranga o \( x \) me \( y \):
\[
x = 3 – 5y
\]
3. Whakaotia te whārite mō \( y \):
\[
\begin{align}
x &= 3 – 5y
x – 3 &= -5y
y &= \frac{3 – x}{5}
\end{whakahāngai}
\]
4. Tuhia te Mahi Whakamuri:
Ko te mahi whakamuri \( g^{-1}(x) \) ko \( g^{-1}(x) = \frac{3 – x}{5} \).
Tauira Pātai 3
Pātai: Mēnā ko \( h(x) = \sqrt{x + 2} \), kimihia te mahi whakamuri \( h^{-1}(x) \).
Kōrero:
1. Whakakapia te \( h(x) \) ki te \( y \):
\[
y = \sqrt{x + 2}
\]
2. Whakawhitihia ngā tūranga o \( x \) me \( y \):
\[
x = \sqrt{y + 2}
\]
3. Whakaotia te whārite mō \( y \):
\[
\begin{align}
x &= \sqrt{y + 2} \\
x^2 &= y + 2 \\
y &= x^2 – 2
\end{whakahāngai}
\]
4. Tuhia te Mahi Whakamuri:
Ko te mahi whakamuri \( h^{-1}(x) \) ko \( h^{-1}(x) = x^2 – 2 \).
Tauira Pātai 4
Pātai: Kimihia te mahi whakamuri o \( k(x) = \ln(x – 1) \) (me \( x > 1 \)).
Kōrero:
1. Whakakapia te \( k(x) \) ki te \( y \):
\[
y = \ln(x – 1)
\]
2. Whakawhitihia ngā tūranga o \( x \) me \( y \):
\[
x = \ln(y – 1)
\]
3. Whakaotia te whārite mō \( y \):
\[
\begin{align}
x &= \ln(y – 1) \\
e^x &= y – 1 \\
y &= e^x + 1
\end{whakahāngai}
\]
4. Tuhia te Mahi Whakamuri:
Ko te mahi whakamuri \( k^{-1}(x) \) ko \( k^{-1}(x) = e^x + 1 \).
Whakamutunga
Ko te mārama ki ngā mahi whakahurihuri me whai mahi whakaharatau me te mārama taahiraa-i-te-taahiraa ki te ariā me ōna whakamahinga. Ko te tukanga matua ko te whakawhiti taurangi, te whakaoti whārite, me te tuhi i te hua whakamutunga hei mahi whakahurihuri. Mā te ako i ngā tauira rapanga maha, pērā i ngā mea kua kōrerohia i runga ake nei, ka āwhina i a tātou ki te whakawhanake ake i ō tātou pūkenga ki te tautuhi me te mārama ki te ariā o ngā mahi whakahurihuri.
Mā te mahi me te māramatanga whānui ki ngā tauira rapanga, ka taea e tātou te whakaoti rapanga rerekē e pā ana ki ngā mahi whakamuri me te māia ake.
