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Operationes fvnctionvm, inversa et transformatio delineationvm

Svmmarivm

Haec lectio fvnctiones considerat tamqvam res qvibvs operationes adhibentvr. Svmmam, prodvctvm, qvotientem et compositionem fvnctionvm realivm cvm earvm dominiis definit, et compositionem commvtativam non esse ostendit. E lectione de copiis, relationibvs et fvnctionibvs associativitatem compositionis, iniectivitatem et theorema fvnctionis inversae svmit, et addit qvae fvnctionibvs realibvs propria svnt: dominivm compositionis, fvnctiones monotonas, inversam fvnctionis stricte monotonae, inversam compvtandam variabilem expediendo et dominivm restringendo, delineationemqve inversae, qvae est reflexio delineationis originalis ad rectam y=xy = x. Deinde fvnctiones pares et impares, translationes, dilatationes et reflexiones delineationis, fvnctionesqve per frvsta definitas et cvm valoribvs absolvtis considerat.

Proposita discendi

  1. Fvnctiones addere, mvltiplicare, dividere et componere, et dominivm cvivsqve eventvs ex dominiis fvnctionvm qvibvs operatio adhibetvr determinare.
  2. Diivdicare vtrvm fvnctio iniectiva, svperiectiva an biiectiva sit, et eivs inversam compvtare, dominivm, vbi opvs est, restringendo.
  3. Symmetrias delineationis agnoscere: ad axem verticalem, ad originem et ad rectam y=xy = x.
  4. Delineationem fvnctionis ex delineatione alterivs per translationes, dilatationes, reflexiones et valores absolvtos obtinere, ordinem servando qvo adhibentvr.
  5. Operationes fvnctionvm et fvnctionem inversam in problematis applicatis interpretari, vt in aeqvatione lentivm, in qva inversa mvnera obiecti et imaginis permvtat.

Operationes fvnctionvm

In lectione de fvnctionibvs algebraicis fvnctio definita est tamqvam regvla qvae cviqve elemento svi dominii vnam prorsvs imaginem assignat, et visvm est fvnctiones algebraicas ex variabili per svmmas, prodvcta, qvotientes et radices formari. Hae operationes intellegi possvnt tamqvam operationes inter fvnctiones: dvae fvnctiones in singvlis pvnctis addvntvr, mvltiplicantvr vel dividvntvr, et altera ad eventvm alterivs adhiberi potest. Qvod si ita est, fvnctiones implicatae considerantvr in simpliciores resolvendo.

Sint ff et gg fvnctiones reales variabilis realis. Earvm svmma f+gf + g, differentia f−gf - g et prodvctvm f⋅gf \cdot g svnt fvnctiones qvae cviqve xx svmmam, differentiam et prodvctvm imaginvm assignant; earvm qvotiens fg\frac{f}{g} cviqve xx qvotientem imaginvm assignat:

f+g:x↦f(x)+g(x),f⋅g:x↦f(x)⋅g(x),fg:x↦f(x)g(x)f + g : x \mapsto f(x) + g(x), \qquad f \cdot g : x \mapsto f(x) \cdot g(x), \qquad \frac{f}{g} : x \mapsto \frac{f(x)}{g(x)}

Vt hae expressiones sensvm habeant, xx ad vtrvmqve dominivm pertinere debet; in qvotiente praeterea imago g(x)g(x) evanescere non potest. Inde dominia operationvm seqvvntvr:

Dom⁡(f+g)=Dom⁡(f)∩Dom⁡(g),Dom⁡(fg)={x∈Dom⁡(f)∩Dom⁡(g):g(x)≠0}\operatorname{Dom}(f + g) = \operatorname{Dom}(f) \cap \operatorname{Dom}(g), \qquad \operatorname{Dom}\left( \frac{f}{g} \right) = \{ x \in \operatorname{Dom}(f) \cap \operatorname{Dom}(g) : g(x) \neq 0 \}

Dominivm differentiae et dominivm prodvcti cvm dominio svmmae congrvvnt, et prodvctvm fvnctionis per nvmervm cc est prodvctvm per fvnctionem constantem cc, cvivs dominivm est R\mathbb{R}. Atqvi dominivm operationis ex fvnctionibvs qvibvs operatio adhibetvr determinatvr, non ex formvla qvae post simplificationem prodit. Si f(x)=x2−1f(x) = x^2 - 1 et g(x)=x−1g(x) = x - 1, qvotiens in toto svo dominio cvm x+1x + 1 congrvit, sed illvd dominivm 11 exclvdit, qvia g(1)=0g(1) = 0: delineatio qvotientis est recta y=x+1y = x + 1 sine pvncto (1,2)(1, 2). Eodem modo, si f(x)=xf(x) = \sqrt{x} et g(x)=−xg(x) = \sqrt{-x}, svmma solvm in x=0x = 0 definita est, vnico pvncto dominiis commvni. Pvncta exclvsa sicvt primvm in lectione de tabvlis signorvm et inaeqvationibvs pvncta evitabilia vocabvntvr, et eorvm effectvs in delineationibvs in lectione de fvnctionibvs rationalibvs considerabitvr.

Compositio fvnctionvm

Gravissima operatio inter fvnctiones nvllvm inter nvmeros simile habet: in eo consistit vt fvnctio ad eventvm alterivs adhibeatvr. Compositio ipsivs ff cvm gg, qvae f∘gf \circ g notatvr et «ff composita cvm gg» legitvr, est fvnctio qvae primvm gg, deinde ff adhibet:

(f∘g)(x)=f(g(x))\left( f \circ g \right)(x) = f(g(x))

Cogitari potest tamqvam series dvarvm machinarvm: xx in gg intrat, qvae g(x)g(x) prodvcit, et hic nvmervs in ff intrat, qvae f(g(x))f(g(x)) prodvcit. Vt series procedat, xx ad dominivm ipsivs gg pertinere debet, et eivs imago ad dominivm ipsivs ff:

Dom⁡(f∘g)={x∈Dom⁡(g):g(x)∈Dom⁡(f)}\operatorname{Dom}(f \circ g) = \{ x \in \operatorname{Dom}(g) : g(x) \in \operatorname{Dom}(f) \}

Animadvertatvr ordo: in f∘gf \circ g prior agit fvnctio dextera scripta. Fvnctiones algebraicae svnt compositiones fvnctionvm simpliciorvm; exempli gratia, 1−x2\sqrt{1 - x^2} est radix qvadrata composita cvm 1−x21 - x^2, et hanc strvctvram agnoscere est qvod eivs dominivm compvtare sinit.

ExemplvmDvae compositiones diversae

Sint f(x)=xf(x) = \sqrt{x} et g(x)=1−x2g(x) = 1 - x^2. Determinare fvnctiones f∘gf \circ g et g∘fg \circ f cvm earvm dominiis, easqve comparare. Figvra qvamqve compositionem repraesentat tamqvam iter per tres rectas nvmericas: rectam ipsivs xx, rectam primae imaginis et rectam secvndae.

Demonstratio

  1. (f∘g)(x)=1−x2,1−x2≥0\left( f \circ g \right)(x) = \sqrt{\resaltar{1 - x^2}}, \quad 1 - x^2 \geq 0

    Dominivm ipsivs gg est R\mathbb{R}, et dominivm ipsivs ff ex realibvs non negativis constat. Vt f∘gf \circ g compvtetvr, imago g(x)g(x) non negativa esse debet; valores ipsivs xx qvi eam negativam facivnt, vt x=2x = 2, per secvndam machinam non transevnt.

  2. (1−x)(1+x)≥0↔Dom⁡(f∘g)=[−1,1]\resaltar{(1 - x)(1 + x)} \geq 0 \leftrightarrow \dato{fg}{\operatorname{Dom}(f \circ g) = \icc{-1}{1}}

    Differentiam qvadratorvm in factores resolvimvs. Per lemma de signo prodvcti ambo factores non negativi esse debent, qvia, vt ambo negativi essent, x>1x > 1 et x<−1x < -1 simvl reqvirerentvr.

  3. (g∘f)(x)=1−(x)2=1−x,Dom⁡(g∘f)=[0,∞)\left( g \circ f \right)(x) = 1 - \resaltar{\left( \sqrt{x} \right)^2} = \resaltar{1 - x}, \quad \dato{gf}{\operatorname{Dom}(g \circ f) = \ico{0}{\infty}}

    Pro g∘fg \circ f nvmervs xx ad dominivm ipsivs ff pertinere debet, et x\sqrt{x} semper ad dominivm ipsivs gg pertinet. Simplificando formvla vestigivm radicis amittit, sed dominivm id servat.

  4. (f∘g)(12)=34=32,(g∘f)(12)=12\left( f \circ g \right)\left( \frac{1}{2} \right) = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}, \quad \left( g \circ f \right)\left( \frac{1}{2} \right) = \frac{1}{2}

    Dvae compositiones svnt fvnctiones diversae: dominia diversa habent, et in eodem pvncto vtriqve commvni, vt x=12x = \frac{1}{2}, valores diversos accipivnt. Ergo compositio commvtativa non est.

Qvamqvam commvtativa non est, compositio associativa est: tribvs fvnctionibvs concatenatis, nihil refert vtrvm primae dvae an postremae dvae privs conivngantvr. Pro fvnctionibvs inter copias datas hoc in lectione de copiis, relationibvs et fvnctionibvs partis primae demonstratvm est. Vnvm fvnctionibvs realibvs proprivm est dominivm, qvod hic non ante declaratvr, sed ex formvlis dedvcitvr. Atqvi, definitione dominii compositionis bis adhibita, nvmervs xx in dominio ipsivs h∘(g∘f)h \circ (g \circ f) est prorsvs cvm est in dominio ipsivs ff, eivs imago f(x)f(x) in dominio ipsivs gg et g(f(x))g(f(x)) in dominio ipsivs hh; eademqve condicio pro (h∘g)∘f(h \circ g) \circ f prodit. Ad hoc dominivm commvne restrictae, fvnctiones concatenatae svnt fvnctiones inter copias, et theorema illivs lectionis in vtroqve casv valorem h(g(f(x)))h(g(f(x))) dat. Ideo simpliciter scribitvr h∘g∘fh \circ g \circ f.

Dom⁡(h∘g∘f)={x∈Dom⁡(f):f(x)∈Dom⁡(g)∧g(f(x))∈Dom⁡(h)}\operatorname{Dom}(h \circ g \circ f) = \{ x \in \operatorname{Dom}(f) : f(x) \in \operatorname{Dom}(g) \wedge g(f(x)) \in \operatorname{Dom}(h) \}