Sketch the graph of $f$ for $-30\le x\le 30$, clearly indicating the points of intersection with each axis and any asymptotes.

With justification, state if each stationary point is a minimum, maximum or horizontal point of inflection.

Find the equation of the horizontal asymptote. 

Find the exact value of $\underset{0}{\overset{1}{\int }}f\left(x\right)dx$, giving the answer in the form $\text{ln}q\text{,}q\in \mathbb{Q}$.

Find the co-ordinates of all stationary points.

Find an expression for ${\theta }_p$ in terms of $p$.

Show that $p=\frac{q^2+q-1}{2q+1}$.

find the coordinates of the $y$-intercept.

show that there are no $x$-intercepts.

sketch the graph, showing clearly any asymptotic behaviour.

Write down the equation of the vertical asymptote.

By sketching the graph of $p$ as a function of $q$, determine the range of values of $p$ for which there are possible values of $q$.

Find the expected number of weeks in the year in which Lucca eats no bananas.

Solve the inequality $f\left(x\right)\ge g\left(x\right)$.

Write down an expression for $f\text{'}\left(x\right)$.

Show that $g^{-1}$ exists.

Consider the graphs of $y=\frac{x^2}{x-3}$ and $y=m\left(x+3\right)$, $m\in \mathbb{R}$.

Find the set of values for $m$ such that the two graphs have no intersection points.

Sketch the graph of $y=f\left(x\right)$ for $-3\le x\le 3$, showing the values of any axes intercepts, the coordinates of any local maxima and local minima, and giving the equations of any asymptotes.

Find the $x$-coordinate of the point of inflexion.

Find the value of $p$ and the value of $q$.

Hence find $g^{-1}\left(x\right)$.

Hence, given that $f^{-1}$ does not exist, show that $b^2-3ac>0$.

Sketch the graph of $y=\frac{1-3x}{x-2}$, showing clearly any asymptotes and stating the coordinates of any points of intersection with the axes.

As $x\to \pm \mathrm{\infty }$ the graph of $f\left(x\right)$ approaches an oblique straight line asymptote.

Divide $2x^2-5x-12$ by $x+2$ to find the equation of this asymptote.

Find the probability that Lucca eats at least one banana in a particular day.

$y$-axis.

Sketch the curve $y=f\left(x\right)$, clearly indicating any asymptotes with their equations. State the coordinates of any local maximum or minimum points and any points of intersection with the coordinate axes.

The oblique asymptote of the graph of $f$ can be written as $y=ax+b$ where $a, b\in \mathrm{\mathbb{Q}}$.

Find the value of $a$ and the value of $b$.

Show that $f\left(x\right)$ has no vertical asymptotes.

$g\left(x\right)$ can be written in the form $p(x-2{)}^3+q$ , where $p, q \in \mathrm{ℝ}$.

Find the value of $p$ and the value of $q$.

Sketch the graphs of $y=g\left(x\right)$ and $y=g^{-1}\left(x\right)$ on the same set of axes, indicating the points where each graph crosses the coordinate axes.

Find an expression for $f\text{'}\left(x\right)$.

Find all the intercepts of the graph of $f\left(x\right)$ with both the $x$ and $y$ axes.

Find the equations of all the asymptotes on the graph of $y=g\left(x\right)$.

State each of the transformations in the order in which they are applied.

By considering the graph of $y=g\left(x\right)-f\left(x\right)$, or otherwise, solve $f\left(x\right)<g\left(x\right)$ for $x\in \mathrm{ℝ}$.

Express $\frac{1}{f\left(x\right)}$ in partial fractions.

Hence find the exact value of $\underset{0}{\overset{3}{\int }}\frac{1}{f\left(x\right)}dx$, expressing your answer as a single logarithm.

$x$-axis.

Write down the equation of the vertical asymptote of the graph of $f$.