Funksiya — zamonaviy matematikaning markaziy tili: bir kattalik o‘zgarganda ikkinchisi qanday o‘zgarishini ifodalaydi. Algebraik formulalar, grafiklar, fizik modellar, iqtisodiy bog‘lanishlar, statistika va keyingi ko‘rsatkichli, logarifmik, trigonometrik mavzularning barchasi funksiya tushunchasiga tayanadi.
Domainning har bir elementiga codomain/range tomondan aynan bitta qiymat mos qo‘yadigan munosabat.
Har input uchun faqat bitta output bo‘lishi kerak.
Misol: $f(x)=2x+1$.
Bu emas: {(1,2),(1,3)} funksiya emas.
💡 Turli inputlar bir xil outputga ega bo‘lishi mumkin.
Ikki to‘plam elementlari orasidagi ordered pairlar to‘plami.
Har relation funksiya bo‘lavermaydi.
Misol: {(1,2),(2,2)}.
Bu emas: Funksiya bo‘lishi uchun bir input ikki outputga ketmasligi kerak.
💡 Function — relationning maxsus turi.
Funksiya qabul qiladigan barcha ruxsat etilgan inputlar to‘plami.
Qaysi x larni kiritish mumkinligini bildiradi.
Misol: $f(x)=1/(x-2)$ uchun $x\ne2$.
Bu emas: Denominator zero inputni domain ichiga olish.
💡 Kontekst domainni algebraik domaindan toraytirishi mumkin.
Funksiya haqiqatan hosil qiladigan barcha outputlar to‘plami.
Qaysi y lar chiqishi mumkin.
Misol: $f(x)=x^2$ uchun $y\ge0$.
Bu emas: Codomain bilan range har doim bir xil deb olish.
💡 Range grafikning vertikal qamrovi.
Input rolidagi o‘zgaruvchi.
Qiymati tanlanadi yoki beriladi.
Misol: $x$ in $y=f(x)$.
Bu emas: Outputni independent deb olish.
💡 Ko‘pincha horizontal axisda.
Inputga bog‘liq holda aniqlanadigan output.
$x$ o‘zgarsa $y=f(x)$ o‘zgaradi.
Misol: $y=3x-2$.
Bu emas: Bir x uchun ikki y berish.
💡 Ko‘pincha vertical axisda.
$f(x)$ — f funksiyaning x inputdagi qiymati.
Bu f ko‘paytirilgan x emas.
Misol: $f(3)$ means input 3.
Bu emas: $f(x)=fx$ deb o‘qish.
💡 Argument qavs ichida.
$(x,y)$ ko‘rinishidagi input-output juftligi.
Birinchi koordinata input, ikkinchisi output.
Misol: $(2,5)$ means f(2)=5.
Bu emas: (5,2) ayni nuqta emas.
💡 Graph point bilan bevosita bog‘liq.
Funksiyadagi barcha $(x,f(x))$ nuqtalar to‘plamining koordinata tekisligidagi tasviri.
Input-output bog‘lanishining vizual ko‘rinishi.
Misol: $y=x^2$ parabola.
Bu emas: Grafikdagi har qanday chiziq funksiya emas.
💡 Vertical line test ishlatiladi.
$f(c)=0$ bo‘ladigan domain elementi c.
Grafik x-axisni kesadigan x qiymat.
Misol: $f(x)=x-4$ noli 4.
Bu emas: y-interceptni zero deb atash.
💡 x-intercept coordinates $(c,0)$.
Grafikning x-axis bilan kesishgan nuqtasi.
Bu yerda y=0.
Misol: $(4,0)$.
Bu emas: x=0 dagi nuqtani x-intercept deyish.
💡 Function zeros bilan bog‘liq.
Grafikning y-axis bilan kesishgan nuqtasi.
Bu yerda x=0.
Misol: $f(0)=b$ → $(0,b)$.
Bu emas: 0 domain’da bo‘lmasa y-intercept bor deyish.
💡 At most one y-intercept for a function.
Domainning turli qismlarida turli formulalar bilan berilgan funksiya.
Qaysi intervalga tushsa, shu qoida ishlaydi.
Misol: $|x|=x$ if $x\ge0$, $-x$ if $x<0$.
Bu emas: Bir x uchun bir vaqtning o‘zida zid ikki qoida.
💡 Boundary conditions muhim.
Intervalda $x_1<x_2$ bo‘lsa $f(x_1)<f(x_2)$ bo‘ladigan funksiya.
Chapdan o‘ngga output ko‘tariladi.
Misol: $f(x)=x^3$ all realda o‘suvchi.
Bu emas: Har qanday positive function o‘suvchi emas.
💡 Strict vs nondecreasing farqlanadi.
Intervalda $x_1<x_2$ bo‘lsa $f(x_1)>f(x_2)$.
Chapdan o‘ngga output pasayadi.
Misol: $f(x)=-x$.
Bu emas: Manfiy qiymatli funksiya avtomatik kamayuvchi emas.
💡 Interval bilan aytiladi.
$f(-x)=f(x)$ bo‘ladigan symmetric-domain funksiya.
y-axisga nisbatan simmetrik.
Misol: $x^2$.
Bu emas: $x^3$ emas.
💡 Domain ham 0 ga nisbatan symmetric bo‘lishi kerak.
$f(-x)=-f(x)$ bo‘ladigan symmetric-domain funksiya.
Origin ga nisbatan 180° simmetrik.
Misol: $x^3$.
Bu emas: $x^2$ emas.
💡 0 domain’da bo‘lsa odd function uchun f(0)=0.
Bir funksiyaning outputini ikkinchi funksiyaga input qilish.
Avval ichki funksiya ishlaydi.
Misol: $(f\circ g)(x)=f(g(x))$.
Bu emas: $fg$ ni composition deb avtomatik o‘qish.
💡 Domain ikki bosqich bilan cheklanadi.
$f(x_1)=f(x_2)$ bo‘lsa $x_1=x_2$ bo‘ladigan funksiya.
Turli inputlar turli output beradi.
Misol: $f(x)=2x+1$.
Bu emas: $x^2$ on all reals emas.
💡 Horizontal line test.
$f^{-1}$ original funksiyaning input va output rollarini qaytaradi va composition identity beradi.
Outputdan inputni tiklaydi.
Misol: $f(x)=2x+3$, $f^{-1}(x)=(x-3)/2$.
Bu emas: $f^{-1}(x)$ ni $1/f(x)$ deb olish.
💡 Original funksiya one-to-one bo‘lishi kerak yoki domain cheklanadi.
Har ruxsat etilgan input aynan bitta outputga aylantiriladi.
$x\mapsto f(x)$
Same input cannot produce two different outputs.
Bir funksiya formula, jadval, ordered pairs, mapping va grafik bilan ifodalanishi mumkin.
$f: x↦y$
Representation o‘zgarsa funksiya o‘zgarmaydi.
Grafikdagi har vertical line grafikni ko‘pi bilan bir nuqtada kessa, relation function.
$x=c$
One x, at most one y.
Denominator 0 emas; even-root radikand ≥0; log argument >0 kabi cheklovlar domain beradi.
$D_f$
Intersect all restrictions.
Real modelda matematik domain physical/semantic ma’no bilan torayishi mumkin.
$D_{context}\subseteq D_{algebra}$
Time, length, count restrictions.
Grafikning vertikal qamrovi range’ni beradi.
$R_f$
Scan bottom-to-top.
f(a) topish uchun formuladagi har x o‘rniga a qo‘yiladi.
$f(a)$
Parentheses preserve structure.
Zeros interval sign changes yoki boundarylar uchun critical points bo‘lishi mumkin.
$f(x)=0$
x-intercepts correspond to real zeros.
Increasing/decreasing/constant behavior x-axis intervali bo‘yicha yoziladi.
$increasing on (a,b)$
Report x-intervals, not y-values.
Local extremum nearby points bilan, absolute extremum butun domain bilan taqqoslanadi.
$f(c)=M$
Endpoint may be absolute extremum.
Ikki nuqta orasida output change/input change.
$\frac{\Delta y}{\Delta x}$
Secant slope.
Har input faqat sharti bajarilgan piece bilan hisoblanadi.
$cases$
Open/closed endpoint symbols matter.
Absolute value sign asosida ikki linear qoida.
$|x|=x\text{ if }x\ge0;\quad |x|=-x\text{ if }x<0$
V-shape.
$f(x)+k$ grafikni k birlik vertikal siljitadi.
$g(x)=f(x)+k$
k>0 up, k<0 down.
$f(x-h)$ grafikni h birlik o‘ngga siljitadi.
$g(x)=f(x-h)$
Inside sign appears reversed.
$-f(x)$ x-axis bo‘yicha; $f(-x)$ y-axis bo‘yicha akslantiradi.
$-f(x), f(-x)$
Output sign vs input sign.
$af(x)$ outputlarni a ga ko‘paytiradi.
$g=af$
|a|>1 stretch; 0<|a|<1 compression; a<0 also reflection.
Sum/difference/product common domain intersectionda; quotientda g(x)≠0 ham kerak.
$f\pm g,fg,f/g$
Domain must be tracked.
$(f\circ g)(x)$ da avval g, keyin f ishlaydi.
$f(g(x))$
Usually f∘g ≠ g∘f.
x g domainida va g(x) f domainida bo‘lishi kerak.
$D_{f\circ g}=\{x\in D_g:g(x)\in D_f\}$
Inner output restriction.
Inverse function bo‘lishi uchun original function outputs takrorlanmasligi kerak.
$f^{-1}$
Horizontal line test.
f va f^{-1} grafiklari y=x ga nisbatan simmetrik.
$(a,b)↔(b,a)$
Domain/range swap.
$x^2$ kabi function domain cheklansa one-to-one bo‘lib inverse olishi mumkin.
$x^2, x\ge0$
Inverse then sqrt(x).
Formula topish yetarli emas: domain, units va predicted behavior real vaziyatga mos bo‘lishi kerak.
$y=f(x)$
Interpret input/output in context.
Output depends on input x.
Shart: x in domain
Same input cannot have two outputs.
Shart: Same function/input
Xususiy holatlar: Converse is one-to-one, not general function.
x-intercept input.
Shart: c in domain
Xususiy holatlar: Graph point (c,0).
Graph at x=0.
Shart: 0 in domain
Xususiy holatlar: Point (0,f(0)).
Secant slope.
Shart: a,b in domain; a≠b
Xususiy holatlar: Linear function gives constant rate.
Average rate over step h.
Shart: h≠0; both inputs in domain
Xususiy holatlar: Derivative precursor.
y-axis symmetry.
Shart: Symmetric domain
Origin symmetry.
Shart: Symmetric domain
Xususiy holatlar: f(0)=0 if 0 in domain.
Different rules on different intervals.
Shart: Pieces define single output
Xususiy holatlar: Boundary overlap must not conflict.
Absolute value as two linear rules.
Shart: x real
Shift graph up/down.
Xususiy holatlar: k>0 up.
Shift graph right/left.
Xususiy holatlar: h>0 right.
Reflect output signs.
Reflect input signs.
Scale all y-values by a.
Shart: a≠0
Xususiy holatlar: a<0 includes x-axis reflection.
Combined graph transformation.
Shart: a,b≠0
Xususiy holatlar: Horizontal scale factor 1/|b|.
Pointwise sum.
Shart: x in D_f∩D_g
Pointwise difference.
Shart: x in D_f∩D_g
Pointwise product.
Shart: x in D_f∩D_g
Pointwise quotient.
Shart: x in D_f∩D_g and g(x)≠0
Xususiy holatlar: Exclude zeros of g.
Apply g then f.
Shart: x in D_g and g(x) in D_f
Xususiy holatlar: Order matters.
Tracks both domain gates.
Injectivity criterion.
Inverse undoes function.
Shart: Proper domains/ranges
Input/output sets swap.
Shart: f one-to-one
Solve y=ax+b for x, then swap labels.
Shart: a≠0
Domain restriction makes x² one-to-one.
Shart: Original domain x≥0
Xususiy holatlar: Range/domain x≥0.
Linear function through origin.
Xususiy holatlar: k=y/x for x≠0.
Tekislikdagi grafik y ni x ning funksiyasi sifatida ifodalashi uchun va faqat shunda har bir vertical line grafikni ko‘pi bilan bir nuqtada kesadi.
Bir x ustida ikki nuqta bo‘lsa ayni inputning ikki outputi bor.
Berilgan: Relation grafigi.
Isbotlash kerak: Vertical line test function ta’rifiga ekvivalentligini ko‘rsatish.
Vertical line test function ta’rifiga ekvivalent. ∎
$(f\circ g)(x)=f(g(x))$ aniqlangan iff x g ning domainida va g(x) f ning domainida.
Ichki machine ishlashi va uning natijasi tashqi machine uchun ruxsat etilishi kerak.
Berilgan: f va g funksiyalar.
Isbotlash kerak: Composite domain formulasini isbotlash.
Composition domain teoremasi isbotlandi. ∎
Funksiya inverse relationi yana funksiya bo‘lishi uchun va faqat shunda original funksiya one-to-one bo‘ladi.
Agar bitta outputga ikki input kelsa, teskari yo‘nalishda bitta inputga ikki output chiqadi.
Berilgan: f funksiya va uning inverse relationi.
Isbotlash kerak: Inverse relation funksiya iff f one-to-one ekanini ko‘rsatish.
Inverse function mavjud iff original function one-to-one. ∎
One-to-one f uchun inverse funksiyaning domaini f ning range’i, range’i esa f ning domainidir.
Inverse input va output rollarini almashtiradi.
Berilgan: f one-to-one va inverse mavjud.
Isbotlash kerak: Domain va range almashishini ko‘rsatish.
Inverse domain va range almashadi. ∎
Symmetric domainli f juft iff grafigi y-axisga nisbatan simmetrik; f toq iff grafigi origin ga nisbatan simmetrik.
x ni -x ga almashtirish reflectionni kodlaydi.
Berilgan: f domaini 0 ga nisbatan symmetric.
Isbotlash kerak: Even/odd algebraic tests graph symmetryga ekvivalentligini ko‘rsatish.
Parity va graph symmetry equivalence isbotlandi. ∎
💡 Maslahat: Har bir birinchi koordinatani tekshiring.
✅ Javob: Ha, funksiya
Nega bu usul ishlaydi: Function definition directly.
⚠️ Turli inputlar bir xil output 5 ga ega bo‘lishi mumkin.
💡 Maslahat: Input 1 ga qarang.
✅ Javob: Yo‘q, funksiya emas
Nega bu usul ishlaydi: Function uniqueness condition.
⚠️ Outputlarning takrorlanishi emas, inputning ikki outputga ketishi muammo.
💡 Maslahat: x=4 ni barcha x lar o‘rniga qo‘ying.
✅ Javob: $21$
Nega bu usul ishlaydi: Function evaluation substitution.
⚠️ Qavslarni saqlang.
💡 Maslahat: Ikki qiymatni alohida yozing.
✅ Javob: $2ah+h^2$
Nega bu usul ishlaydi: Symbolic evaluation.
⚠️ $(a+h)^2=a^2+h^2$ emas.
💡 Maslahat: Denominator 0 bo‘lmasin.
✅ Javob: $(-\infty,5)\cup(5,\infty)$
Nega bu usul ishlaydi: Rational-domain rule.
⚠️ x=5 ni kiritmang.
💡 Maslahat: Radikand ≥0.
✅ Javob: $[3,\infty)$
Nega bu usul ishlaydi: Even-root domain.
⚠️ Radikandni >0 qilish shart emas; 0 mumkin.
💡 Maslahat: Ikkala restrictionni kesishiring.
✅ Javob: $[-2,1)\cup(1,\infty)$
Nega bu usul ishlaydi: All algebraic restrictions intersect.
⚠️ Faqat root restrictionni olish yetarli emas.
💡 Maslahat: $x^2\ge0$.
✅ Javob: $[3,\infty)$
Nega bu usul ishlaydi: Minimum from nonnegative square.
⚠️ Domain bilan range’ni almashtirmang.
💡 Maslahat: Principal root nonnegative.
✅ Javob: $[0,\infty)$
Nega bu usul ishlaydi: Principal-root range.
⚠️ Domain [2,∞) — range emas.
💡 Maslahat: $f(x)=0$ qo‘ying.
✅ Javob: $2,3$
Nega bu usul ishlaydi: Zeros solve f(x)=0.
⚠️ x-intercepts nuqtalar (2,0),(3,0).
💡 Maslahat: $x=0$.
✅ Javob: $(0,2)$
Nega bu usul ishlaydi: y-intercept is f(0).
⚠️ x-intercept bilan adashtirmang.
💡 Maslahat: Vertical line test.
✅ Javob: Yo‘q
Nega bu usul ishlaydi: Vertical line test fails.
⚠️ Yuqori yarim aylana alohida function bo‘lishi mumkin.
💡 Maslahat: Vertex x=2.
✅ Javob: O‘suvchi $(-\infty,2)$; kamayuvchi $(2,\infty)$
Nega bu usul ishlaydi: Parabola behavior around vertex.
⚠️ Intervallarni y qiymatlar bilan yozmang.
💡 Maslahat: Square nonnegative.
✅ Javob: Minimum qiymat $-4$, x=-1 da
Nega bu usul ishlaydi: Vertex/minimum.
⚠️ Minimum x=-1 emas; x joylashuv, -4 qiymat.
💡 Maslahat: Secant slope formula.
✅ Javob: $5$
Nega bu usul ishlaydi: Average change per unit input.
⚠️ $(16-1)/4$ deb bo‘lmang.
💡 Maslahat: Boundary qaysi piece’ga kiradi?
✅ Javob: $3$
Nega bu usul ishlaydi: Piecewise condition determines rule.
⚠️ x<1 piece’dan foydalanmang.
💡 Maslahat: Ichidagi ifoda ishorasini ajrating.
✅ Javob: $x-3$ for $x\ge3$; $3-x$ for $x<3$
Nega bu usul ishlaydi: Absolute value definition.
⚠️ Boundary x=3 faqat bitta branchga yetarli.
💡 Maslahat: Outside +4.
✅ Javob: 4 birlik yuqoriga
Nega bu usul ishlaydi: Vertical translation.
⚠️ O‘ngga 4 emas.
💡 Maslahat: Inside x-5.
✅ Javob: 5 birlik o‘ngga
Nega bu usul ishlaydi: Horizontal translation sign reversal.
⚠️ Chapga 5 deb o‘ylamang.
💡 Maslahat: Output sign flips.
✅ Javob: x-axisga nisbatan aks
Nega bu usul ishlaydi: -f(x) reflection.
⚠️ f(-x) bilan adashtirmang.
💡 Maslahat: General form $a f(x-h)+k$.
✅ Javob: O‘ngga 3, x-axis reflection, vertikal ×2, yuqoriga 1
Nega bu usul ishlaydi: General transformation parameters.
⚠️ Inside/outside parameterlarni adashtirmang.
💡 Maslahat: Common domain intersection.
✅ Javob: $\sqrt{x}+1/(x-4)$, domain $[0,4)\cup(4,\infty)$
Nega bu usul ishlaydi: Function operation uses shared domain.
⚠️ Sum formula domainni avtomatik all real qilmaydi.
💡 Maslahat: Quotientda g(x)≠0.
✅ Javob: $(x+1)/(x-2)$, $x\ne2$
Nega bu usul ishlaydi: Quotient denominator restriction.
⚠️ f va g alohida all-real bo‘lsa ham quotient x=2 da yo‘q.
💡 Maslahat: Avval g(3).
✅ Javob: $19$
Nega bu usul ishlaydi: Composition order: inner then outer.
⚠️ f(3) va g(3) ni ko‘paytirmang.
💡 Maslahat: g(x) ni f inputiga qo‘ying.
✅ Javob: $2x^2-7$
Nega bu usul ishlaydi: Direct substitution.
⚠️ g(f(x)) boshqa funksiya.
💡 Maslahat: Inner domain va outer forbidden input.
✅ Javob: $[1,10)\cup(10,\infty)$
Nega bu usul ishlaydi: Composite domain has two gates.
⚠️ Faqat x≥1 ni olish yetarli emas.
💡 Maslahat: Ikkalasini alohida hisoblang.
✅ Javob: $f\circ g=2x+1$, $g\circ f=2x+2$
Nega bu usul ishlaydi: Composition generally noncommutative.
⚠️ Tartibni almashtirmang.
💡 Maslahat: $f(a)=f(b)$ dan boshlang.
✅ Javob: Ha, one-to-one
Nega bu usul ishlaydi: Injectivity algebraic criterion.
⚠️ Nonzero-slope linear functions one-to-one.
💡 Maslahat: f(1) va f(-1).
✅ Javob: Yo‘q
Nega bu usul ishlaydi: Counterexample is enough.
⚠️ Function bo‘lish bilan one-to-one bo‘lish boshqa.
💡 Maslahat: y=x² dan x ni nonnegative branchda yeching.
✅ Javob: $f^{-1}(x)=\sqrt{x}$, $x\ge0$
Nega bu usul ishlaydi: Domain restriction removes ± ambiguity.
⚠️ Inverse ±sqrt(x) emas; inverse funksiya bitta output beradi.
💡 Maslahat: y=5x-2 ni x ga yeching.
✅ Javob: $f^{-1}(x)=\frac{x+2}{5}$
Nega bu usul ishlaydi: Solve and swap.
⚠️ $1/(5x-2)$ inverse emas.
💡 Maslahat: Ikkala composition identity.
✅ Javob: $g=f^{-1}$
Nega bu usul ishlaydi: Two composition identities verify inverse.
⚠️ Faqat bitta random qiymatni tekshirish umumiy proof emas.
💡 Maslahat: y=1/(x-2) ni x ga yeching.
✅ Javob: $f^{-1}(x)=2+1/x$; domain $x\ne0$; range $y\ne2$
Nega bu usul ishlaydi: Inverse swaps domain/range.
⚠️ Algebraik formula bilan domainni alohida yozing.
💡 Maslahat: f(-x) ni hisoblang.
✅ Javob: Juft
Nega bu usul ishlaydi: Even test.
⚠️ Faqat darajalar juft ekanini ko‘rish bu polynomialda shortcut, lekin test universal.
💡 Maslahat: f(-x).
✅ Javob: Toq
Nega bu usul ishlaydi: Odd test.
⚠️ Odd polynomial terms origin symmetry beradi.
💡 Maslahat: f(-x) ni f(x) va -f(x) bilan solishtiring.
✅ Javob: Na juft, na toq
Nega bu usul ishlaydi: Parity definitions.
⚠️ Har function juft yoki toq bo‘lishi shart emas.
💡 Maslahat: $y=kx$.
✅ Javob: $f(x)=\frac52x$
Nega bu usul ishlaydi: Direct variation constant from one pair.
⚠️ Intercept qo‘shmang; direct variation origin orqali o‘tadi.
💡 Maslahat: Fixed + variable cost.
✅ Javob: $C(x)=10000+2500x$
Nega bu usul ishlaydi: Real situation becomes linear function.
⚠️ x — km, C — so‘m ekanini yozing.
💡 Maslahat: Masofa manfiy emas.
✅ Javob: Kontekstual domain $[0,\infty)$
Nega bu usul ishlaydi: Context can restrict algebraic domain.
⚠️ Formula all-real bo‘lsa real model ham all-real degani emas.
💡 Maslahat: Differences constant.
✅ Javob: Rate $3$, $f(x)=3x+2$
Nega bu usul ishlaydi: Constant first difference identifies linear function.
⚠️ Outputlarni x bilan adashtirmang.
💡 Maslahat: 15000 ikkinchi piece’da.
✅ Javob: $2000$
Nega bu usul ishlaydi: Piecewise real-world model.
⚠️ Ikkinchi bracketdagi faqat ortiqcha qism 20% bilan olinadi.
💡 Maslahat: Inside x+3, outside ×2 and -1.
✅ Javob: Chapga 3, vertikal ×2, pastga 1; domain $[-3,\infty)$
Nega bu usul ishlaydi: Transformation preserves parent structure with updated domain.
⚠️ x+3 ni o‘ngga 3 deb o‘qmang.
💡 Maslahat: f(x)=g(x).
✅ Javob: $(-1,1)$ va $(3,9)$
Nega bu usul ishlaydi: Graph intersections solve equal outputs.
⚠️ Faqat x qiymatlarni berib nuqta so‘ralganini unutmayin.
💡 Maslahat: Reciprocal term 0 bo‘la olmaydi.
✅ Javob: $(-\infty,3)\cup(3,\infty)$
Nega bu usul ishlaydi: Shifted reciprocal misses horizontal asymptote value.
⚠️ Domain x≠2 bilan range y≠3 ni adashtirmang.
💡 Maslahat: Initial value va exponential term limitini ko‘ring.
✅ Javob: Boshlang‘ich $80^\circ$, limit $25^\circ$, domain $[0,\infty)$
Nega bu usul ishlaydi: Function features have direct context meanings.
⚠️ Algebraic formula negative t da ham mavjud bo‘lishi kontekstda uni ruxsat etmaydi.
❌ $f(x)$ ni $f\cdot x$ deb o‘qish.
Function notation multiplication emas.
✅ $f(x)$ ni “f ning x dagi qiymati” deb o‘qing.
$f(3)$ — input 3.
❌ Bitta inputga ikki output bo‘lsa ham function deyish.
Function definition buziladi.
✅ Har input uchun aynan bitta outputni tekshiring.
(1,2),(1,4) function emas.
❌ Output takrorlansa function emas deb o‘ylash.
Turli inputlar bir outputga borishi mumkin.
✅ Input uniqueness muhim.
(1,5),(2,5) function.
❌ Domain va range’ni almashtirish.
Domain inputlar, range outputlar.
✅ Horizontal coverage = domain, vertical coverage = range.
$x^2$: D=R, range [0,∞).
❌ Rational function domainida denominator zero nuqtani qoldirish.
Division by zero undefined.
✅ Denominator zerosni chiqarib tashlang.
$1/(x-2)$: x≠2.
❌ Even root domainida radikandni >0 qilish.
Root of 0 defined.
✅ Numerator/ordinary root uchun radikand ≥0.
$\sqrt{x-3}$ da x=3 kiradi.
❌ y-intercept uchun f(x)=0 yechish.
Bu zeros/x-intercepts beradi.
✅ y-intercept uchun x=0 qo‘ying.
Point (0,f(0)).
❌ O‘sish intervalini y qiymatlar bilan yozish.
Increasing/decreasing input interval bo‘yicha aytiladi.
✅ x-axis intervalini yozing.
(-∞,2), not y<5.
❌ Average rate of change’da denominatorni b yoki a deb olish.
Input change b-a bo‘lishi kerak.
✅ $[f(b)-f(a)]/(b-a)$.
x² on [1,4] →5.
❌ Piecewise boundaryda noto‘g‘ri piece tanlash.
< va ≤ endpoint inclusionni belgilaydi.
✅ Shartni aynan tekshiring.
x=1 uchun x≥1 branch.
❌ $f(x-h)$ ni chapga h deb talqin qilish.
Horizontal shifts ichki sign bilan teskari ko‘rinadi.
✅ $f(x-h)$ → right h.
$|x-5|$ right 5.
❌ $-f(x)$ va $f(-x)$ ni adashtirish.
Birinchisi output, ikkinchisi input signini o‘zgartiradi.
✅ -f(x): x-axis; f(-x): y-axis reflection.
$-\sqrt{x}$ vs $\sqrt{-x}$.
❌ Function sum/product domainini avtomatik barcha real olish.
Har operand o‘z domain restrictioniga ega.
✅ Domain intersectionni oling.
$\sqrt{x}+1/(x-4)$.
❌ Function quotientda g(x)=0 ni unutish.
Quotient undefined.
✅ Shared domain + denominator function nonzero.
(f/g)(2) invalid if g(2)=0.
❌ Composition tartibini almashtirish.
$f(g(x))$ va $g(f(x))$ odatda boshqa.
✅ Avval qavs ichidagi functionni bajaring.
f=x+1,g=2x →2x+1 vs2x+2.
❌ Composite domain’da faqat inner domainni tekshirish.
Inner output outer domain tashqarisiga tushishi mumkin.
✅ x∈D_g va g(x)∈D_f.
$1/(\sqrt{x-1}-3)$ da x=10 chiqariladi.
❌ $f^{-1}(x)$ ni $1/f(x)$ deb olish.
Inverse function reciprocal emas.
✅ y=f(x) ni x ga yechib input/outputni almashtiring.
5x-2 inverse (x+2)/5.
❌ One-to-one bo‘lmagan functionga inverse function yozish.
Inverse relation vertical-line testni buzadi.
✅ Domainni cheklang yoki inverse function yo‘q deb ayting.
$x^2$ all realda inverse function emas.
❌ Inverse’da domain va range’ni eski holicha qoldirish.
Input/output rollari almashadi.
✅ $D_{f^{-1}}=R_f$, $R_{f^{-1}}=D_f$.
$1/(x-2)$ inverse domain x≠0.
❌ Real modelda algebraik domainni kontekst domainsiz qabul qilish.
Negative time/length kabi values matematik mavjud bo‘lsa ham ma’nosiz.
✅ Context constraintsni final domain bilan kesishiring.
Taxi distance x≥0.
Har equation y ni x ning funksiyasi qiladi.
Aylana kabi relation bir x ga ikki y berishi mumkin; vertical line test kerak.
Funksiya bo‘lish uchun barcha outputlar turli bo‘lishi kerak.
Bu one-to-one sharti; oddiy functionda output takrorlanishi mumkin.
Domain doim barcha real sonlar.
Formula va kontekst denominator, roots, logs va real meaning orqali domainni cheklaydi.
Range ni formula ko‘rib domain kabi topish mumkin.
Range output behaviorni talab qiladi; graph, inverse reasoning yoki extrema kerak bo‘lishi mumkin.
Grafik yuqorida bo‘lsa function o‘suvchi.
Positive value va increasing behavior boshqa tushunchalar.
$f(x)+k$ va $f(x+k)$ bir xil siljish.
Birinchisi vertical, ikkinchisi horizontal shift.
Composition kommutativ: f∘g=g∘f.
Odatda noto‘g‘ri; order matters.
Har function inverse functionga ega.
Faqat one-to-one function inverse relationi function bo‘ladi; aks holda domain restriction kerak.
Inverse graph x-axisga akslanadi.
Inverse graph y=x chizig‘iga nisbatan akslanadi.
Real-life function model formulasi matematik jihatdan aniqlangan barcha x uchun ishlaydi.
Model domaini fizik/semantic shartlar bilan cheklanadi.
Position-time, velocity-time, temperature va energy bog‘lanishlari funksiyalar bilan ifodalanadi.
Cost, revenue, profit, demand va tax piecewise/linear/nonlinear functions sifatida model qilinadi.
Feature→prediction mapping, transformations va model response functions.
Input-output systems, calibration curves, transfer relations va operating domains.
Population growth, dose-response va time-dependent processes.
Function abstraction, composition va input/output contracts matematik funksiya g‘oyasiga parallel.
Area/volume parametrga bog‘liq funksiya bo‘lib, domain va extrema analysis talab qiladi.
Grafik o‘qish, domain/range, inverse va transformations SAT hamda kirish imtihonlarida asosiy ko‘nikma.
Funksiya mavzusida avval function gate va domain/range aniqlanadi; keyin graph features, transformations/operations yoki inverse/model yo‘liga o‘tiladi. Yakunda composite domain, one-to-one va kontekst cheklovlari tekshiriladi.
Cheat sheet: function = each input has exactly one output; $y=f(x)$; domain = allowed inputs; range = attained outputs; zero iff $f(x)=0$; y-intercept = $f(0)$ if 0 is in domain; vertical line test checks function; horizontal line test checks one-to-one; average rate of change = $(f(b)-f(a))/(b-a)$; $(f\circ g)(x)=f(g(x))$; inverse satisfies $f^{-1}(f(x))=x$ on the proper domain; inverse swaps domain/range; graph transforms: $f(x)+k$ up, $f(x-h)$ right, $-f(x)$ reflect across x-axis, $f(-x)$ reflect across y-axis.
Keyingi ko‘rsatkichli, logarifmik va trigonometrik mavzularda aynan shu domain, range, monotonicity, transformation, composition va inverse tushunchalari ishlatiladi. Logarifmik funksiya eksponensial funksiyaning inverse’i sifatida shu mavzuning bevosita davomidir.
Oldin bilishingiz kerak: Chiziqli funksiya, uning grafigi va xossalari, Kvadrat funksiya, Matnli masalalar
Bog'liq mavzular: Arifmetik kvadrat ildiz va uning xossalari, Modulli ifodalar va tenglamalar
Keyingi mavzular: Ko'rsatkichli tenglama, Logarifmik funksiya va uning xossalari, Trigonometriya. Asosiy tushunchalar