MATHHX B

MATHHX B

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(numerical range)}\TextOrMath { }{\ }}\) \(\def \LWRsiunitxdecimal {.}\)

3.6 Ekstra til lineære funktioner ()

Dette afsnit indeholder nogle lidt sværere øvelser i den forstand, at det ikke er ”typeopgaver”, så man skal tænke lidt. Hvis du elsker at bruge din hjerne, er dette afsnit guf.

Øvelse 3.6.1

Lad \(f\) være en begrænset lineær funktion, som opfylder:

  • \(\Dm (f)=]2;6]\)

  • \(\Vm (f)=[-2;8[\)

  • \(f\) er aftagende.

  • a) Bestem ved beregning en forskrift for \(f\).

 3.6.1

  • a) \(f(x)=-\frac {5}{2}x+13\qquad ,\qquad 2<x\leq 6\)

    Huskede du begrænsningen på \(x\)?

Øvelse 3.6.2

Grafen for en lineær funktion \(f\) går igennem punktet \((3,13)\) og skærer \(y\)-aksen i \(-14\).

  • a) Bestem ved beregning en forskrift for \(f\).

 3.6.2

  • a) \(f(x)=9x-14\).

Øvelse 3.6.3

En lineær funktion \(f\) har nulpunkt i \(x=3\) og en hældning på \(\frac {1}{2}\).

  • a) Bestem ved beregning en forskrift for \(f\).

 3.6.3

  • a) \(f(x)=\frac {1}{2}x-\frac {3}{2}\).

Øvelse 3.6.4

Lad \(f(x)=2x-3\) og \(g(x)=-x+b\).

  • a) Beregn \(b\), så førstekoordinaten til skæringspunktet mellem graferne for \(f\) og \(g\) er \(4\).

 3.6.4

  • a) \(b=9\)

Øvelse 3.6.5

Lad \(f(x)=-3x+1\). Grafen for \(f\) går igennem punktet \((x_0,4)\).

  • a) Beregn \(x_0\).

 3.6.5

  • a) \(x_0=-1\)

Øvelse 3.6.6

Lad \(f(x)=ax+b\) være en lineær funktion, hvor \(a\neq 0\).

  • a) Udled en formel for nulpunktet for \(f\). Dvs. find frem til en formel, som kan bruges til at regne nulpunktet ud fra \(a\) og \(b\).

 3.6.6

  • a) \(x=-\frac {b}{a}\)

Øvelse 3.6.7

I denne opgave skal vi se på et fiktivt skattesystem til indkomstskat. Reglerne skal være:

  • De første \(100.000\) kr. af indkomsten beskattes ikke.

  • Resten af indkomsten beskattes med \(60\%\) (uuuuhhh).

  • a) En person tjener \(80.000\) kr. Hvor meget skal der betales i skat?

  • b) En person tjener \(140.000\) kr. Hvor meget skal der betales i skat?

  • c) Opstil en funktion, som beskriver skattebeløbet som funktion af indtægten.

 3.6.7

  • a) \(0\) kr.

  • b) \(24.000\)

  • c) \(f(x) = \begin {cases} 0 & \text {for } x\leq 100000 \\ 0{,}6x-60000 & \text {for } x> 100000 \end {cases}\)

Øvelse 3.6.8

Fahrenheit (\(\degree \)F) er en enhed for temperatur. Man bruger den i USA, da de elsker skøre måleenheder derovre.

Fahrenheit er oprindelig defineret ud fra frysepunktet for en bestemt saltopløsning og ”almindelig kropstemperatur”, men det glemmer vi lige nu. I stedet vil vi tage udgangspunkt i, at vand fryser ved \(32\,\degree \)F og koger ved \(212\,\degree \)F. Hvis du ikke allerede ved det, kan jeg oplyse at vand fryser ved \(\SI {0}{\celsius }\) og koger ved \(\SI {100}{\celsius }\).

  • a) Opstil forskriften for en lineær funktion, der kan bruges til at omregne fra Celsius til Fahrenheit.

  • b) Opstil forskriften for en lineær funktion, der kan bruges til at omregne fra Fahrenheit til Celsius. Brug gerne brøker i forskriften.

  • c) Jeg har hørt en tommelfingerregel, som går ud på, at man kan omregne fra Fahrenheit til Celsius ved at ”trække \(30\) fra og dele med \(2\)”. Er det korrekt?

 3.6.8

  • a) \(f(x)=1{,}8x+32\), hvor \(x\) er temperaturen i Celsius, og \(f(x)\) er temperaturen i Fahrenheit.

  • b) \(f(x)=\frac {5}{9}x-\frac {160}{9}\), hvor \(x\) er temperaturen i Fahrenheit, og \(f(x)\) er temperaturen i Celsius. Forskriften kan også skrives som \(f(x)=0{,}56x-17{,}78\).

  • c) Hvis man har en temperatur \(x\) i Fahrenheit og bruger tommelfingerreglen, får man

    \[\frac {x-30}{2}=\frac {1}{2}x-15\]

    Dette svarer ikke helt til \(f(x)=\frac {5}{9}x-\frac {160}{9}\), men lad os prøve et eksempel. Har man \(90\degree \) Fahrenheit og bruger tommelfingerreglen får man:

    \[\frac {90-30}{2}=30\]

    Så \(90\,\degree \)F svarer altså til \(\SI {30}{\celsius }\) ifølge regelen. Bruger man den præcise forskrift, får man:

    \[f(90)=\frac {5}{9}\cdot 90 - \frac {160}{9}=\frac {290}{9}\approx 32\]

    Sååå lidt derhen af... Godt nok til at finde ud af om det er t-shirt eller lange ærmer, I guess.