MATHHX B
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2.1 Introduktion til funktioner
En funktion er sammenknytning af to mængder. Det skal forstås på den måde, at til hvert element i den første mængde knyttes netop ét element i den anden mængde. Det er nemmest at forstå med et eksempel:
Diagrammet ovenover viser en funktion, som til hvert tal knytter det dobbelte tal. Til \(1\) knyttes \(2\), til \(2\) knyttes \(4\) osv. Vi kalder tallene i den første mængde for \(x\)’er (dvs. \(1,2,3\) osv.) og tallene i den anden
mængde er \(y\)’er (\(2,4,6\) osv.). Vi kalder også \(y\)’erne for funktionsværdier. Funktionens navn er \(f\).
Øvelse 2.1.1
I denne øvelse tages udgangspunkt i den konkrete funktion, som er beskrevet ovenover.
-
a) Hvilket tal knytter \(f\) til tallet \(5\)?
-
b) Hvis \(x=4\), hvad er så \(y\)?
-
c) Hvad er funktionsværdien når \(x=-3\)?
-
d) Hvis \(y=20\), hvad var så \(x\)?
Forskrift
Vi beskriver ofte funktioner vha. en forskrift. Tager vi den før omtalte funktion, kan vi beskrive den med forskriften \(f(x)=2x\). Her er \(f\) funktions navn, og \(2x\) betyder, at vi skal gange hvert \(x\) med \(2\) for
at få den tilhørende funktionsværdi (dvs. \(y\)-værdien).
Læg mærke til, at der står \(f(15)=30\) i stedet for \(y=30\). Det er smart, da man så kan se, at man har fat i den funktionsværdi, der hører til \(15\). Udtrykket \(f(x)\) læses ”f af x” og \(f(30)\) læses ”f af 30”.
Øvelse 2.1.2
Vi bliver ved den samme funktion \(f(x)=2x\).
Når vi skal opskrive en funktion, vil vi normalt ikke lave et diagram, vi vil nøjes med angive forskriften.
Nogle gange hedder funktionen ikke \(f\). Den kunne også hedde \(g\), eller \(h\). Kun fantasien sætter grænser her.
Øvelse 2.1.3
Betragt funktion \(g(x)=2x-1\).
-
a) Bestem funktionsværdien, der hører til \(x\)-værdien \(3\).
-
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 2.1.4
Betragt diagrammerne:
-
a) Bestem en forskrift, som passer med diagrammet for funktionen \(f\).
-
b) Bestem en forskrift, som passer med diagrammet for funktionen \(h\).
Løsning 2.1.4
-
a) \(f(x)=4x\)
-
b) \(h(x)=x+1\)
Når vi har en funktion, kalder vi \(x\) for den uafhængige variabel, fordi vi selv kan bestemme, hvilken \(x\)-værdi vi putter ind i funktionen. Vi kalder \(y\) for den afhængige variabel, da den jo afhænger af den \(x\)-værdi, vi
putter 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? Det
er dog meget nemt:
Øvelse 2.1.5
Bestem funktionsværdierne
Beregning af \(x\) ud fra \(y\)
Har man en forskrift \(f(x)\), og en funktionsværdi \(y\), kan man finde \(x\)-værdien ved at sætte \(y\) ind i stedet for \(f(x)\) i forskriften.
-
Eksempel 2.1.4
Lad \(f(x)=4x-1\). Vi vil gerne finde den \(x\)-værdi, som hører til funktionsværdien \(7\). Vi sætter \(7\) ind i stedet for ”\(f(x)\)” og løser ligningen:
\(\seteqnumber{0}{2.}{0}\)
\begin{align*}
f(x) & =4x-1\\ 7 & =4x-1\\ 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 2.1.6
Lad \(f(x)=-3x+1\).
Løsning 2.1.6
-
a) \(x=-3\)
-
b) \(x=\frac {1}{3}\)
Øvelse 2.1.7 (Svær)
Lad \(f(x)=x^2\).
Løsning 2.1.7
-
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 på HHX, så det er vel
begrænset, hvad jeg kan forvente. Du lærer at løse den slags ligninger (og nogle som er endnu sværere) i afsnittet om polynomier. Peace.