# Oppgavesamling: Rasjonale funksjoner :::::::::::::::{admonition} Oppgave 1 --- class: problem-level-1 --- I {numref}`fig-rasjonale-funksjoner-oppgavesamling-oppgave-1` vises grafen til en rasjonal funksjon $f$. En horisontal og en vertikal asymptote for $f$ er tegnet inn. :::{figure} ./figurer/oppgave_1/graf.svg --- name: fig-rasjonale-funksjoner-oppgavesamling-oppgave-1 width: 100% class: no-click, adaptive-figure --- viser grafen til en rasjonal funksjon $f$. En horisontal og en vertikal asymptote for $f$ er tegnet inn. ::: ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a Bestem et mulig uttrykk for $f(x)$. ::::{admonition} Fasit --- class: answer, dropdown --- $$ f(x) = \dfrac{-2x + 6}{x + 1} $$ :::: ::::::::::::: :::::::::::::{tab-item} b Løs ulikheten $f(x) < 0$. ::::{admonition} Fasit --- class: answer, dropdown --- $$ x \in \mathbb{R} \setminus [-1, 3] $$ :::: ::::::::::::: :::::::::::::{tab-item} c Løs ulikheten $f(x) \geq 2$. ::::{admonition} Fasit --- class: answer, dropdown --- $$ x \in \langle -1, 1] $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{admonition} Oppgave 2 --- class: problem-level-1 --- En rasjonal funksjon $f$ er gitt ved $$ f(x) = \dfrac{-x + 2}{x - 1} $$ ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a Løs likningen $f(x) = 0$. ::::{admonition} Fasit --- class: answer, dropdown --- $$ x = 2. $$ :::: ::::::::::::: :::::::::::::{tab-item} b Avgjør om $f$ har en vertikal asymptote, og bestem likningen til asymptoten hvis den finnes. ::::{admonition} Fasit --- class: answer, dropdown --- $$ x = 1. $$ :::: ::::::::::::: :::::::::::::{tab-item} c Avgjør om $f$ har en horisontal asymptote og bestem likningen til asymptoten hvis den finnes. ::::{admonition} Fasit --- class: answer, dropdown --- $$ y = -1. $$ :::: ::::::::::::: :::::::::::::{tab-item} d Lag en skisse av grafen til $f$. ::::{admonition} Fasit --- class: answer, dropdown --- :::{figure} ./figurer/oppgave_2/d.svg --- width: 100% class: no-click, adaptive-figure --- ::: :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{admonition} Oppgave 3 --- class: problem-level-1 --- I {numref}`fig-rasjonale-funksjoner-oppgavesamling-oppgave-3` vises grafen til en rasjonal funksjon $f$. En horisontal og en vertikal asymptote for $f$ er tegnet inn. :::{figure} ./figurer/oppgave_3/graf.svg --- name: fig-rasjonale-funksjoner-oppgavesamling-oppgave-3 width: 100% class: no-click, adaptive-figure --- viser grafen til en rasjonal funksjon $f$. En horisontal og en vertikal asymptote for $f$ er tegnet inn. ::: ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a Bestem et mulig uttrykk for $f(x)$. ::::{admonition} Fasit --- class: answer, dropdown --- $$ f(x) = \dfrac{3x + 3}{x - 2} $$ :::: ::::::::::::: :::::::::::::{tab-item} b Bestem definisjonsmengden til $f$. ::::{admonition} Fasit --- class: answer, dropdown --- $$ D_f = \mathbb{R} \setminus \{2\} $$ :::: ::::::::::::: :::::::::::::{tab-item} c Løs $f(x) > 0$. ::::{admonition} Fasit --- class: answer, dropdown --- $$ x \in \mathbb{R} \setminus [-1, 2]. $$ :::: ::::::::::::: :::::::::::::{tab-item} d Løs $f(x) \leq 2$. ::::{admonition} Fasit --- class: answer, dropdown --- $$ x \in [-7, 2\rangle . $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{admonition} Oppgave 4 --- class: problem-level-1 --- En rasjonal funksjon $f$ er gitt ved $$ f(x) = \dfrac{2x - 1}{3x + 4} $$ ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a Avgjør om $f$ har nullpunkter og bestem nullpunktene hvis de finnes. ::::{admonition} Fasit --- class: answer, dropdown --- $$ x = \dfrac{1}{2} $$ :::: ::::::::::::: :::::::::::::{tab-item} b Avgjør om $f$ har vertikale asymptoter og bestem likningene til asymptotene hvis de finnes. ::::{admonition} Fasit --- class: answer, dropdown --- $$ x = -\dfrac{4}{3} $$ :::: ::::::::::::: :::::::::::::{tab-item} c Avgjør om $f$ har horisontale asymptoter og bestem likningene til asymptotene hvis de finnes. ::::{admonition} Fasit --- class: answer, dropdown --- $$ y = \dfrac{2}{3} $$ :::: ::::::::::::: :::::::::::::{tab-item} d Lag en skisse av grafen til $f$. ::::{admonition} Fasit --- class: answer, dropdown --- :::{figure} ./figurer/oppgave_4/d.svg --- width: 100% class: no-click, adaptive-figure --- ::: :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{admonition} Oppgave 5 --- class: problem-level-2 --- To rasjonale funksjoner $f$ og $g$ er gitt ved $$ f(x) = \dfrac{x + 2}{(x - 3)^2} \quad \text{og} \quad g(x) = \dfrac{(x + 2)^2}{x - 3} $$ Nedenfor vises fire grafer der én av dem er grafen til $f$ og én av dem er grafen til $g$. ::::{multi-plot2} --- rows: 2 cols: 2 xmin: -10 xmax: 10 ymin: -10 ymax: 10 fontsize: 25 lw: 3.5 --- :::{plot} width: 100% function: (x + 2) / (x - 3)**2 ticks: off text: 5, 8, "A", center-center, bbox hline: 0, dashed, red vline: 3, dashed, red ::: :::{plot} width: 100% function: -(x + 2) / (x - 3)**2 ticks: off text: 5, 8, "B", center-center, bbox hline: 0, dashed, red vline: 3, dashed, red ::: :::{plot} width: 100% function: (x + 2)**2 / (x - 3) ticks: off text: 5, 35, "C", center-center, bbox ymin: -40 ymax: 40 line: 1, 7, dashed, red vline: 3, dashed, red ::: :::{plot} width: 100% function: (x - 2)**2 / (x + 3) ticks: off text: 5, 35, "D", center-center, bbox ymin: -40 ymax: 40 line: 1, -7, dashed, red vline: -3, dashed, red ::: :::: ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a Avgjør hvilken figur som viser grafen til $f$. ::::{admonition} Fasit --- class: answer, dropdown --- Graf **A**. :::: ::::::::::::: :::::::::::::{tab-item} b Avgjør hvilken figur som viser grafen til $g$. ::::{admonition} Fasit --- class: answer, dropdown --- Graf **C**. :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{admonition} Oppgave 6 --- class: problem-level-2 --- Tre rasjonale funksjoner $f$, $g$ og $h$ er gitt ved $$ f(x) = \dfrac{x^2 + 4x - 5}{x^2 - 9} \quad\quad g(x) = \dfrac{x^2 - 1}{x^2 - 9} \quad\quad h(x) = \dfrac{x^2 - 9}{x^2 - 1} $$ Avgjør hvilke av figurene nedenfor som viser grafene til $f$, $g$ og $h$. ::::{multi-plot2} --- rows: 3 cols: 2 xmin: -10 xmax: 10 ymin: -10 ymax: 10 fontsize: 25 lw: 3.5 --- :::{plot} function: (x**2 - 1) / (x**2 - 9) ticks: off hline: 1, dashed, red vline: -3, dashed, red vline: 3, dashed, red text: 5, 8, "A", center-center, bbox ::: :::{plot} function: -(x**2 - 1) / (x**2 - 9) ticks: off hline: 1, dashed, red vline: -3, dashed, red vline: 3, dashed, red text: 5, 8, "B", center-center, bbox ::: :::{plot} function: (x**2 - 9) / (x**2 - 1) ticks: off hline: 1, dashed, red vline: -1, dashed, red vline: 1, dashed, red text: 5, 8, "C", center-center, bbox ymin: -15 ymax: 15 ::: :::{plot} function: -(x**2 - 9) / (x**2 - 1) ticks: off hline: -1, dashed, red vline: -1, dashed, red vline: 1, dashed, red text: 5, 8, "D", center-center, bbox ymin: -15 ymax: 15 ::: :::{plot} function: -(x**2 + 4*x - 5) / (x**2 - 9) ticks: off hline: -1, dashed, red vline: -3, dashed, red vline: 3, dashed, red text: 5, 8, "E", center-center, bbox ::: :::{plot} function: (x**2 + 4*x - 5) / (x**2 - 9) ticks: off hline: 1, dashed, red vline: -3, dashed, red vline: 3, dashed, red text: 5, 8, "F", center-center, bbox ::: :::: ::::{admonition} Fasit --- class: answer, dropdown --- * Graf **F** viser $f$. * Graf **A** viser $g$. * Graf **C** viser $h$. :::: ::::::::::::::: --- :::::::::::::::{admonition} Oppgave 7 --- class: problem-level-2 --- En rasjonal funksjon $f$ er gitt ved $$ f(x) = \dfrac{x^2 - 4}{(x + 2)(x - 4)} $$ ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a Bestem nullpunktene til $f$, dersom de finnes. ::::{admonition} Fasit --- class: answer, dropdown --- $$ x = 2. $$ :::: ::::::::::::: :::::::::::::{tab-item} b Bestem likningene til de vertikale asymptotene til $f$, dersom de finnes. ::::{admonition} Fasit --- class: answer, dropdown --- $$ x = 4. $$ :::: ::::::::::::: :::::::::::::{tab-item} c Bestem likningen til $f$ sin skrå eller horisontale asymptote, dersom den finnes. ::::{admonition} Fasit --- class: answer, dropdown --- $$ y = 1. $$ :::: ::::::::::::: :::::::::::::{tab-item} d Løs ulikheten $f(x) > 0$ ::::{admonition} Fasit --- class: answer, dropdown --- $$ x \in \langle \gets, 2 \rangle \cup \langle 4, \to \rangle \setminus \{-2\} $$ :::: ::::::::::::: :::::::::::::{tab-item} e Lag en skisse av grafen til $f$. ::::{admonition} Fasit --- class: answer, dropdown --- :::{figure} ./figurer/oppgave_7/e.svg --- width: 100% class: no-click, adaptive-figure --- ::: :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{admonition} Oppgave 8 --- class: problem-level-2 --- En rasjonal funksjon $g$ er gitt ved $$ g(x) = \dfrac{x^2 + 6x + 9}{x - 2} $$ ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a Bestem nullpunktene til $g$, dersom de finnes. ::::{admonition} Fasit --- class: answer, dropdown --- $$ x = -3. $$ :::: ::::::::::::: :::::::::::::{tab-item} b Bestem likningene til $g$ sine vertikale asymptoter, dersom de finnes. ::::{admonition} Fasit --- class: answer, dropdown --- $$ x = 2. $$ :::: ::::::::::::: :::::::::::::{tab-item} c Bestem likningen til en eventuell skrå eller horisontal asymptote til $g$. ::::{admonition} Fasit --- class: answer, dropdown --- $$ y = x + 8. $$ :::: ::::::::::::: :::::::::::::{tab-item} d Løs ulikheten $g(x) < 0$. ::::{admonition} Fasit --- class: answer, dropdown --- $$ x \in \langle \gets, 2 \rangle \setminus \{-3\} $$ :::: ::::::::::::: :::::::::::::{tab-item} e Lag en skisse av grafen til $g$. ::::{admonition} Fasit --- class: answer, dropdown --- :::{figure} ./figurer/oppgave_8/e.svg --- width: 100% class: no-click, adaptive-figure --- ::: :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{admonition} Oppgave 9 --- class: problem-level-2 --- En rasjonal funksjon $f$ er gitt ved $$ f(x) = \dfrac{x^3 - 4x}{(x + 2)(x - 1)} $$ ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a Bestem nullpunktene til $f$, dersom de finnes. ::::{admonition} Fasit --- class: answer, dropdown --- $$ x = 0 \or x = 2. $$ :::: ::::::::::::: :::::::::::::{tab-item} b Bestem likningene til $f$ sine vertikale asymptoter, dersom de finnes. ::::{admonition} Fasit --- class: answer, dropdown --- $$ x = 1. $$ :::: ::::::::::::: :::::::::::::{tab-item} c Bestem likningen til en eventuell skrå eller horisontal asymptote til $f$. ::::{admonition} Fasit --- class: answer, dropdown --- $$ y = x - 1. $$ :::: ::::::::::::: :::::::::::::{tab-item} d Løs ulikheten $f(x) \geq 0$. ::::{admonition} Fasit --- class: answer, dropdown --- $$ x \in [0, 1 \rangle \cup [2, \to \rangle $$ :::: ::::::::::::: :::::::::::::{tab-item} e Lag en skisse av grafen til $f$. ::::{admonition} Fasit --- class: answer, dropdown --- :::{figure} ./figurer/oppgave_9/e.svg --- width: 100% class: no-click, adaptive-figure --- ::: :::: ::::::::::::: :::::::::::::: :::::::::::::::