IB Mathematics Analysis and Approaches Topic AHL…

Consider the rational function $f(x)=\frac{6 x^{2}-5 x-4}{2 x^{2}-8}$ , where $x \in \mathbb{R}, x \neq \pm 2$ . The graph of $y=f(x)$ has a local minimum at point P and a local maximum at point Q.

Express $f(x)$ in the form $\frac{a(x-b)(x-c)}{(x-d)(x-e)}$ , where .

Consider the function defined by

f(x) = \dfrac{7}{x^2 + 6x - 7}, \qquad x \in \mathbb{R},\ x \neq -7,\ x \neq 1.

Consider $f(x)=\frac{x^3+4x^2+x+\lambda}{x^2 - x - 6},\qquad x\in\mathbb{R}\setminus\{-2,3\},\ \lambda\in\mathbb{R}\setminus\{-6, -66\}.$

Define $f(x)=\frac{x-1}{kx+2},\quad k\in\mathbb R,\ x\ne -\frac{2}{k}\ (k\ne0).$

Given $f(x)=\frac{mx+n}{px-q},\quad mq+np\ne0,\quad x\ne \frac{q}{p}.$

Determine the equations of the vertical and horizontal asymptotes of the graph of $y=f(x)$ .

Find the coordinates of the points where the graph of $y=f(x)$ intersects the $x$ -axis and $y$ -axis.

Using a graphical calculator, find the coordinates of the local minimum at P and the local maximum at Q, and hence state the range of $f(x)$ .

Solve the inequality $\frac{6|x|^{2}-5|x|-4}{2|x|^{2}-8} \geq \frac{3}{2}$ , for $x \in \mathbb{R}, x \neq \pm 2$ .

Sketch the graph of $y = f(x)$ , showing the values of any axes intercepts, the coordinates of any local maxima or minima, and the equations of any asymptotes.

The function $g$ is defined by $g(x) = \dfrac{7}{x^2 + 6x - 7}$ for $x \in \mathbb{R}$ , $x > 1$ .

Show that $g^{-1}(x) = -3 + \sqrt{\dfrac{16x + 7}{x}}$ .

State the domain of $g^{-1}$ .

The function $h$ is defined by $h(x) = \arccos\!\left(\dfrac{x}{7}\right)$ .

Show the vertical asymptotes are independent of $\lambda$ and find them.

Write $f(x)=q(x)+\dfrac{r(x)}{x^2-x-6}$ with $q$ and $r$ are linear expressions.

Deduce the oblique asymptote.

Find the intersection point(s) of the graph with its oblique asymptote.

Solve $f(x)\ge x+5$ for $\lambda=-30$ .

Compute $f^{-1}(x)$ and its domain.

Determine for which $k$ values $f$ is self-inverse.

Let $g_k(x)=\dfrac{kx+1}{x-k}$ . Prove self-inverse and find fixed points.

For $k>0$ , solve $g_k(x)\ge x$ . For $k = 2$ , solve to 3 s.f., noting the number of roots.

Show that $f$ is a self-inverse function if and only if $m-q=0$ .

Let $g_s(x)=\dfrac{s x+1}{x-s}$ where . Prove that is a self-inverse function and find its fixed points.

For $s>0$ , solve the inequality $g_s(x)\ge x$ .

IB Mathematics Analysis and Approaches AHL 2.13—rational Functions Questionbank

IB Mathematics Analysis and Approaches (AA) Topic AHL 2.13—rational Functions Questionbank