2.13 Rational functions questions for IB Maths AA

Solve the inequality.

Hence solve ${\displaystyle \frac{3}{\mathrm{\mid }x\mathrm{\mid }-2}}>\mathrm{\mid }x\mathrm{\mid }$, where $x\in \mathbb{R}$, $\mathrm{\mid }x\mathrm{\mid }\mathrm{\ne }2$.

Write down the equation of the vertical asymptote of the curve $y=f(x)$.

Find the equation of the oblique asymptote of the graph of $y=f(x)$.

Solve the inequality $f(x)\le x+4$.

Consider the function $f_1(x)={\displaystyle \frac{x-3}{(x-1)(x-2)}}$, for $x\mathrm{\ne }1,2$.

Write down the equations of the asymptotes of the graph of $y=f_1(x)$.

Use your GDC to find the coordinates of the local maximum point and the local minimum point on the graph of $y=f_1(x)$.

Write down the product of the $y$-coordinates of these two points.

Consider the function $f_2(x)={\displaystyle \frac{x-1.5}{(x-1)(x-2)}}$, for $x\mathrm{\ne }1,2$.

Use your GDC to sketch the graph of $y=f_2(x)$. Hence write down the number of stationary points on the graph and the range of $f_2$.

Make a conjecture about the position of $a$, relative to $p$ and $q$, for which the graph of $y=f(x)$ has stationary points.

Show that $f^{\mathrm{\prime }}(x)=-{\displaystyle \frac{x^2-2ax+a(p+q)-pq}{(x-p{)}^2(x-q{)}^2}}$.

Hence prove your conjecture from part (c).

For the remainder of this question, suppose that $a<p$ or $a>q$, and let $y_1$ and $y_2$ be the $y$-coordinates of the two stationary points.

It can be shown that $y_1$ and $y_2$ are the values of $k$ for which the equation $f(x)=k$ has a repeated root in $x$.

Show that $y_1$ and $y_2$ are the roots of the equation $(p-q{)}^2{k}^2+2(p+q-2a)k+1=0$.

Hence show that $y_1{y}_2$ does not depend on $a$, and explain how this is consistent with your answer to part (a)(iii).

Write down an expression for $y_1+y_2$ in terms of $a$, $p$ and $q$.

The graph of $y={\displaystyle \frac{x-a}{x^2-4}}$, where $a>2$, has a local minimum point with $y$-coordinate ${\displaystyle \frac{1}{2}}$.

Find the value of $a$ and the $y$-coordinate of the local maximum point.

Write down the equation of the vertical asymptote and the equation of the horizontal asymptote of the graph of $y=f(x)$.

Find the coordinates of the points where the graph of $y=f(x)$ meets the axes.

Sketch the graph of $y=f(x)$, showing the asymptotes and the points found in part (b).

Find the $x$-coordinates of the points where the graph of $y=f(x)$ meets the line $y=2x-3$.

Hence solve the inequality $f(x)\ge 2x-3$.

Solve the inequality $\mid f(x)\mid <3$.

Write down the equation of the horizontal asymptote of the graph of $y=f(x)$.

Find the coordinates of the local maximum point and the local minimum point on the graph of $y=f(x)$.

The line $y=1$ meets the graph of $y=f(x)$ at two points.

Find the area of the region enclosed by the graph of $y=f(x)$ and the line $y=1$.

Express $f(x)$ in partial fractions.

Write down the equations of the asymptotes of the graph of $y=f(x)$.

Find the coordinates of the points where the graph of $y=f(x)$ meets the axes.

Use your answer to part (a) to show that $f$ is decreasing on each interval of its domain.

The curve $y=f(x)$ and the two coordinate axes bound a region $R$. Find the area of $R$.

Show that, for every non-zero real number $k$, the equation $f(x)=k$ has exactly two solutions.