MATHHX B
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2.1 Funktioner og forskrifter
Lad os sige, at du har et fritidsjob, hvor du tjener 100 kr. i timen. I denne situation vil der være en sammenhæng mellem det antal timer, du arbejder, og din løn. Sammenhængen kan illustreres med et diagram:
Til venstre i diagrammet ses det antal timer, du arbejder. Disse tal kaldes \(x\)’er. Til højre ses lønnen, som betegnes med \(y\). Sammenhængen mellem \(x\)- og \(y\)-værdierne kaldes en funktion, og den er her betegnet med \(f\). Diagrammet er selvfølgelig ufuldstændigt i den forstand, at det kun viser, hvad der sker, når man arbejder 1, 2 eller 3 timer, men man kan selvfølgelig også arbejde f.eks.
7 timer og tjene 700 kr.
Funktioners \(y\)-værdier kaldes også funktionsværdier.
Øvelse 2.1.1
I denne øvelse tages udgangspunkt i funktionen \(f\) fra diagrammet.
-
a) Hvilken \(y\)-værdi knytter \(f\) til \(x\)-værdien \(2\)?
-
b) Hvad er funktionsværdien, når \(x=13\)?
-
c) Hvilken \(x\)-værdi hører sammen med \(y\)-værdien \(900\)?
Forskrifter
Vi beskriver for det meste funktioner ved hjælp af det, vi kalder en forskrift. Forskriften for en funktion fortæller os, hvordan funktionsværdien regnes ud fra \(x\)-værdien. Tager vi
funktionen, som knyttede løn til arbejdstimer, kan den beskrives med forskriften:
\[ f(x)=100x \]
Ud fra forskriften regner man funktionsværdien, som er betegnet med \(f(x)\) i forskriften, ved at erstatte \(x\) med den konkrete \(x\)-værdi.
-
Eksempel 2.1.1
Vi vil regne funktionsværdien hørende til \(x=15\) for funktionen:
\[ f(x)=100x \]
Vi erstatter \(x\) med \(15\) i forskriften:
\[ f(15)=100\cdot 15=1500 \]
Vi har altså, at \(f(15)=1500\), hvilket betyder, at funktionsværdien er \(1500\), når \(x=15\). Læg mærke til, at vi har erstattet alle \(x\)’erne i forskriften med \(15\), også det \(x\), som er i \(f(x)\).
Øvelse 2.1.2
Vi bliver ved den samme funktion \(f(x)=100x\).
Skrivemåden \(f(x)\) læses ”f af x” og \(f(0)\) læses ”f af nul”.
Funktioner behøver ikke hedde \(f\). De kan også hedde f.eks. \(g\) eller \(h\).
Øvelse 2.1.4
Lad \(g(x)=2x-1\).
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a) Bestem funktionsværdien, der hører til \(x\)-værdien \(3\).
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b) Regn \(g(6)\), \(g(0)\) og \(g(-1)\).
-
c) Ved at prøve dig frem skal du finde den \(x\)-værdi, som giver funktionsværdien \(7\).
Øvelse 2.1.5
Betragt diagrammerne:
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a) Bestem en forskrift, som passer med diagrammet for \(f\).
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b) Bestem en forskrift, som passer med diagrammet for \(g\).
2.1.5
-
a) \(f(x)=4x\)
-
b) \(g(x)=x+1\)
Når vi har en funktion, kalder vi \(x\) for den uafhængige variabel, fordi vi selv kan bestemme, hvilken \(x\)-værdi vi sætter ind i funktionen. Vi kalder \(y\) for den afhængige
variabel, da den jo afhænger af den \(x\)-værdi, vi sætter ind i funktionen.
Konstante funktioner
En konstant funktion er en funktion, som f.eks. \(f(x)=-53\). Mange bliver forvirrede, når de møder konstante funktioner første gang – for hvordan skal man regne funktionsværdierne, når der ikke er noget \(x\) i forskriften?
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Eksempel 2.1.3
Lad \(f(x)=-53\). Vi bestemmer funktionsværdien \(f(6)\) ved at sætte \(6\) ind på \(x\)’ets plads. Det bliver vi hurtigt færdig med, da der ikke er noget \(x\) i forskriften, så
\(f(6)=-53\). Alle funktionsværdier for denne funktion vil på den måde blive \(-53\), og det er derfor, det kaldes en konstant funktion. Funktionsværdien afhænger ikke af \(x\), men er konstant.
Øvelse 2.1.6
Lad \(f(x)=4\).
Beregning af \(x\) ud fra \(y\)
Har man en forskrift \(f(x)\) og en funktionsværdi \(y\), kan man finde \(x\)-værdien ved at sætte \(y\)-værdien ind i stedet for \(f(x)\) i forskriften.
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Eksempel 2.1.4
Lad \(f(x)=4x-1\). Vi bestemmer den \(x\)-værdi, som hører til funktionsværdien \(7\):
\(\seteqnumber{0}{2.}{0}\)
\begin{align*}
f(x) & =4x-1 && (\text {forskrift skrevet op})\\ 7 & =4x-1 && (7\text { indsat i stedet for }f(x))\\ 8 &=4x\\ x & = 2
\end{align*}
At sætte \(7\) ind i stedet for \(f(x)\), som vi gjorde i eksemplet, kaldes også at løse ligningen \(f(x)=7\).
Øvelse 2.1.7
Lad \(f(x)=-3x+1\).
2.1.7
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a) \(x=-3\)
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b) \(x=\frac {1}{3}\)
Øvelse 2.1.8 ()
Lad \(f(x)=x^2\).
2.1.8
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a) Der er to løsninger: \(x=3\) og \(x=-3\). Fandt du dem begge to? Nej, det tænkte jeg nok, men du er jo også kun lige startet, så det er helt ok! Du
lærer at løse den slags ligninger (og nogle som er endnu sværere) i afsnittet om polynomier.