Piv txwv ntawm Cov Lus Nug thiab Kev Sib Tham txog Algebraic Functions
Cov functions algebraic yog ib qho tseem ceeb hauv kev kawm lej, feem ntau tshwm sim hauv kev xeem hauv tsev kawm ntawv thiab kev sib tw lej. Kev nkag siab txog lub tswv yim ntawm cov functions algebra thiab yuav daws cov teeb meem cuam tshuam li cas yog qhov tseem ceeb rau kev paub txog cov ncauj lus no. Tsab xov xwm no yuav piav qhia txog ntau qhov teeb meem piv txwv thiab tham txog cov functions algebraic kom ntxaws.
Pendahuluuan
Ib qho kev ua haujlwm yog ib qho kev sib raug zoo uas txuas txhua lub ntsiab lus hauv ib pawg (hu ua thaj chaw) rau ib qho ntsiab lus hauv lwm pawg (hu ua codomain). Hauv lej, ib qho kev ua haujlwm tuaj yeem qhia ua \(f: A \to B \), qhov twg \(f \) yog ib qho kev ua haujlwm uas txuas cov ntsiab lus hauv pawg \(A \) rau cov ntsiab lus hauv pawg \(B \). Lub cim dav dav rau ib qho kev ua haujlwm yog \(f(x) \), uas txhais tau tias \(f \) yog ib qho kev ua haujlwm uas nyob ntawm qhov hloov pauv \(x \).
Piv txwv lus nug 1: Linear Function
Lus Nug: Txheeb xyuas qhov sib npaug ntawm txoj kab \( f(x) \) uas hla dhau qhov taw tes (2, 3) thiab muaj qhov sib txawv ntawm 4.
Kev Sib Tham:
Tus linear function dav dav muaj daim ntawv \( f(x) = mx + c\), qhov twg \( m\) yog qhov gradient thiab \( c\) yog y-intercept.
1. Hloov tus nqi gradient \( m = 4 \) rau hauv kab zauv:
\[
f(x) = 4x + c
\]
2. Siv qhov chaw (2, 3) los nrhiav \( c \):
\[
3 = 4(2) + c
\]
\[
3 = 8 + c
\]
\[
c = 3 – 8
\]
\[
c = -5
\]
3. Nrog \( m = 4 \) thiab \( c = -5 \), tus qauv ntawm txoj kab yog:
\[
f(x) = 4x – 5
\]
Piv txwv lus nug 2: Quadratic Functions
Lo lus nug: Muab ib qho quadratic function \( f(x) = ax^2 + bx + c \). Yog tias daim duab ntawm qhov function hla dhau cov ntsiab lus (1, 4), (2, 7), thiab (3, 12), txiav txim siab cov nqi ntawm \( a \), \( b \), thiab \( c \).
Kev Sib Tham:
1. Hloov lub ntsiab lus (1, 4) rau hauv kab zauv:
\[
4 = a(1)^2 + b(1) + c
\]
\[
4 = a + b + c \quad \text{(Equation 1)}
\]
2. Hloov lub ntsiab lus (2, 7) rau hauv kab zauv:
\[
7 = a(2)^2 + b(2) + c
\]
\[
7 = 4a + 2b + c \quad \text{(Equation 2)}
\]
3. Hloov lub ntsiab lus (3, 12) rau hauv kab zauv:
\[
12 = a(3)^2 + b(3) + c
\]
\[
12 = 9a + 3b + c \quad \text{(Equation 3)}
\]
4. Daws qhov system ntawm cov kab zauv linear:
- Rho tawm kab zauv 1 ntawm kab zauv 2:
\[
(7 – 4) = (4a + 2b + c) – (a + b + c)
\]
\[
3 = 3a + b \quad \text{(Equation 4)}
\]
- Rho tawm kab zauv 2 ntawm kab zauv 3:
\[
(12 - 7) = (9a + 3b + c) - (4a + 2b + c)
\]
\[
5 = 5a + b \quad \text{(Equation 5)}
\]
5. Rho tus lej 4 ntawm tus lej 5:
\[
(5-3) = (5a + b) - (3a + b)
\]
\[
2 = 2a
\]
\[
ua = 1
\]
6. Hloov \( a = 1 \) rau hauv kab zauv 4:
\[
3 = 3(1) + b
\]
\[
3 = 3 + b
\]
\[
b = 0 hli
\]
7. Hloov \( a = 1 \) thiab \( b = 0 \) rau hauv kab zauv 1:
\[
4 = 1 + 0 + c
\]
\[
c = 3
\]
Yog li, cov nqi ntawm \(a\), \(b\), thiab \(c\) yog:
\[
a = 1, b = 0, c = 3
\]
Yog li, qhov kev ua haujlwm quadratic yog:
\[
f(x) = x^2 + 3
\]
Piv txwv lus nug 3: Functions thiab Trigonometry
Teeb Meem: Muab ib qho kev ua haujlwm \( f(x) = 2 \sin (x) + \cos (x) \). Txheeb xyuas \( f\left(\frac{\pi}{2}\right) \).
Kev Sib Tham:
1. Hloov \( x = \frac{\pi}{2} \) rau hauv qhov kev ua haujlwm:
\[
f\left(\frac{\pi}{2}\right) = 2\sin \left(\frac{\pi}{2}\right) + \cos \left(\frac{\pi}{2}\right)
\]
2. Nco ntsoov tias cov nqi trigonometric:
\[
\sin\left(\frac{\pi}{2}\right) = 1 \quad \text{and} \quad \cos\left(\frac{\pi}{2}\right) = 0
\]
3. Ces peb tau txais:
\[
f\left(\frac{\pi}{2}\right) = 2(1) + 0
\]
\[
f\left(\frac{\pi}{2}\right) = 2
\]
Piv txwv lus nug 4: Kev sib xyaw ua ke ntawm cov haujlwm
Teeb Meem: Muab cov haujlwm \( f(x) = 2x + 1 \) thiab \( g(x) = x^2 - 3 \). Txheeb xyuas \( (f \circ g)(x) \) thiab \( (g \circ f)(x) \).
Kev Sib Tham:
1. \( (f \circ g)(x) \) :
\[
(f \circ g)(x) = f(g(x))
\]
Hloov \( g(x) \) rau hauv \( f(x) \):
\[
g(x) = x^2 – 3
\]
\[
f(g(x)) = f(x^2 – 3)
\]
Siv \( f(x) = 2x + 1 \):
\[
f(x^2 – 3) = 2(x^2 – 3) + 1
\]
\[
= 2x^2 – 6 + 1
\]
\[
= 2x^2 – 5
\]
2. \( (g \circ f)(x) \) :
\[
(g \circ f)(x) = g(f(x))
\]
Hloov \( f(x) \) rau hauv \( g(x) \):
\[
f(x) = 2x + 1
\]
\[
g(f(x)) = g(2x + 1)
\]
Siv \( g(x) = x^2 – 3 \):
\[
g(2x + 1) = (2x + 1)^2 – 3
\]
\[
= 4x^2 + 4x + 1 – 3
\]
\[
= 4x^2 + 4x – 2
\]
Yog li, qhov kawg tshwm sim:
\[
(f \circ g)(x) = 2x^2 – 5
\]
\[
(g \circ f)(x) = 4x^2 + 4x – 2
\]
Xaus
Cov algebraic functions muaj ntau yam, los ntawm linear functions mus rau quadratic functions mus rau function compositions. Tsab xov xwm no nthuav qhia ntau qhov teeb meem piv txwv nrog rau kev sib tham ntxaws ntxaws. Kev nkag siab txog yuav ua li cas daws cov teeb meem no yuav muaj txiaj ntsig zoo rau kev paub txog lub ntsiab lus ntawm algebraic functions thiab kev siv lwm yam kev xav lej.
Yog koj xyaum ua tas li thiab nkag siab zoo txog cov ntsiab lus, kev daws teeb meem algebraic function yuav dhau los ua ib qho kev txawj ntse uas ntseeg tau. Xyaum ua ntxiv thiab tsis txhob yig nrhiav lwm cov peev txheej los pab kom koj nkag siab tob dua.
