MATHHX B
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3.4 Stykkevis lineære funktioner
En stykkevis lineær funktion er en funktion, hvis graf er sammensat af linjestykker.
Da en stykkevis lineær funktion består af flere dele, består forskriften også af flere dele. Funktionen fra eksempel 3.4.1 har forskriften:
\[ f(x) = \begin {cases} -\frac {1}{2}x-1 & \text {for } x<2 \\ 2x-8 & \text {for } x\geq 2 \end {cases} \]
En sådan forskrift kaldes en gaffelforskrift. Vi ser at funktionsværdierne er bestemt ved \(-\frac {1}{2}x-1\) når \(x<2\), og \(2x-8\) når \(x\geq 2\). Vi kan regne \(f(5)\) ved først at observere at \(5\geq 2\),
så vi skal bruge udtrykket \(2x-8\) til regne funktionsværdien. Vi har derfor at
\[f(5)=2\cdot 5-8=2\]
Så \(f(5)=2\).
Øvelse 3.4.1
Vi fortsætter med funktionen med forskriften:
\[ f(x) = \begin {cases} -\frac {1}{2}x-1 & \text {for } x<2 \\ 2x-8 & \text {for } x\geq 2 \end {cases} \]
Beregn
-
a) \(f(-2)\)
-
b) \(f(2)\)
-
c) \(f(3)\)
Løsning 3.4.1
-
a) \(f(-2)=0\)
-
b) \(f(2)=-4\)
-
c) \(f(3)=-2\)
Man kan tegne grafen for en stykkevis lineær funktion ved at tegne alle linjerne i forskriften, hvorefter man visker dele af dem ud. Til sidst kan man tilføje endepunkter. Lad os se på, hvordan man tegner grafen for funktionen:
\[ f(x) = \begin {cases} -\frac {1}{2}x-1 & \text {for } x<2 \\ 2x-8 & \text {for } x\geq 2 \end {cases} \]
Vi starter med at tegne linjerne \(y=-\frac {1}{2}x-1\) og \(y=2x-8\):
Funktionen skal skifte ved \(x=2\), så vi visker grafen for \(y=2x-8\) ud til venstre for \(x=2\) og visker grafen for \(y=-\frac {1}{2}x-1\) ud til højre for \(x=2\).
Vi mangler nu bare at tilføje endepunkter. Da venstre del gælder for \(x<2\), og højre del gælder for \(x\geq 2\), må det være højre del, som gælder for \(x=2\), og derfor skal der være et lukket (udfyldt) endepunkt på højre
del og et åbent endepunkt (ikke udfyldt) på venstre del:
Øvelse 3.4.2
Tegn grafen for funktionerne:
-
a)
\[ f(x) = \begin {cases} 2x+5 & \text {for } x\leq -1 \\ x-4 & \text {for } x > -1 \end {cases} \]
-
b)
\[ g(x) = \begin {cases} x+5 & \text {for } x\leq -2 \\ 3 & \text {for } -2\leq x \leq 1 \\ 4x-3 & \text {for } 1 < x < 2 \\ \end {cases} \]
Løsning 3.4.2
-
a)
-
b)
-
Eksempel 3.4.2
Vi vil bestemme forskriften for den stykkevise lineære funktion givet ved grafen:
Venstre del er nem, det er jo bare den konstante funktion \(f(x)=-3\), den er gældende når \(x\) ligger mellem \(-4\) og \(1\), begge punkter er inklusive, da de er markeret med en udfyldt bolle. Så vi har en gren som hedder:
\[ f(x) = \begin {cases} -3 & \text {for } 4\leq x\leq 1 \\ \ \\ \end {cases} \]
Vi kan bestemme forskriften for højre del ved at vælge to punkter på grafen og så bruge sætning 3.2.1 (den sætning med forskriften ud fra to
punkter). Vi aflæser to pæne punkter på højre del:
\[(2,3)\qquad \text {og}\qquad (4,0)\]
Vi indsætter først i formlen for \(a\):
\(\seteqnumber{0}{3.}{0}\)
\begin{align*}
a & = \frac {y_1-y_0}{x_1-x_0}\\ & = \frac {0-3}{4-2}\\ & =-1{,}5
\end{align*}
Så beregnes \(b\):
\(\seteqnumber{0}{3.}{0}\)
\begin{align*}
b & = y_0-ax_0 \\ & =3-(-1{,}5)\cdot 2\\ & =6
\end{align*}
Altså højre del er bestemt ved \(y=-1{,}5x+6\), og da den er gældende for \(x>0\) (endepunktet er markeret med en ikke udfyldt bolle) får vi:
\[ f(x) = \begin {cases} -3 & \text {for } -4\leq x\leq 1 \\ -1{,}5x+6 & \text {for } x > 1 \\ \end {cases} \]
Øvelse 3.4.3
Betragt den stykkevise lineære funktion \(f\):
Løsning 3.4.3
-
a)
\[ f(x) = \begin {cases} 1{,}5x+5 & \text {for } x\leq -2 \\ -x & \text {for } -2< x \leq 1 \\ 2 & \text {for } 1 <x< 4 \\ \end {cases} \]
-
b) \(f(-2)=2\) og \(f(1)=-1\).
Anvendelse af stykkevis lineære funktioner
Til slut vil vi se på en enkelt anvendelse af stykkevis lineære funktioner. I et progressivt skattesystem betaler dem med højere indkomster en større procentdel i skal end dem med lavere indkomster. Jeg har opgivet at følge med i det
danske skattesystem, men på et tidspunkt (måske 2020?) så det ca. sådan her ud:
Funktionen viser det samlede skattebeløb man betaler som funktion af indkomsten \(x\). Man kan se at kurven har et skarpt knæk ved ca. 50.000 kr. og hvis man kigger nøje, kan man se, at den også knækker ved 520.000 kr.
Øvelse 3.4.4
Betragt grafen for skattesystemet.
-
a) Forklar betydningen af de forskellige hældningerne på grafen.
-
b) Bestem hældningen for den sidste del af grafen (bare sådan ca.) og forklar, hvad den udtrykker.
Løsning 3.4.4
-
a) Hældningerne viser hvor meget man betaler i skat pr. tjente krone.
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b) Sidste del har en hældning på ca. \(0{,}56\). Dette betyder, at man højst kan komme til at betale \(0{,}56\) kr. i skat pr. ekstra krone man tjener.
Denne skat kaldes også ”marginalskatten”, og den er meget slem, synes de rige.