Let $f(x) = x - 8$, $g(x) = x^4 - 3$, and $h(x) = f(g(x))$.

The function $f$ is defined for all $x \in \mathbb{R}$. The line with equation $y = 6x - 1$ is the tangent to the graph of $f$ at $x = 4$.

The function $g$ is defined for all $x \in \mathbb{R}$ where $g(x) = x^2 - 3x$ and $h(x) = f(g(x))$.

SpeedWay airline flies from city $A$ to city $B$. The flight time is normally distributed with a mean of 260 minutes and a standard deviation of 15 minutes. A flight is considered late if it takes longer than 275 minutes.

The flight is considered to be on time if it takes between $m$ and 275 minutes. The probability that a flight is on time is 0.830.

During a week, SpeedWay has 12 flights from city $A$ to city $B$. The time taken for any flight is independent of the time taken by any other flight.

Let $f(x) = x \cdot e^{-x}$ and $g(x) = -3f(x) + 1$.

The graphs of $f$ and $g$ intersect at $x = p$ and $x = q$, where $p < q$.

The function $f$ is defined by $f(x)=\frac{2x+4}{3-x}$, where $x \in \mathbb{R}$, $x \neq 3$.

Write down the equation of

Find the coordinates where the graph of $f$ crosses

A function $f$ is defined by $f(x) = \frac{2x-1}{x+1}$, where $x \in \mathbb{R}$, $x \neq -1$.

The graph of $y = f(x)$ has a vertical asymptote and a horizontal asymptote.

The cubic equation $x^3 - kx^2 + 3k = 0$ where $k > 0$ has roots $\alpha, \beta$ and $\alpha + \beta$.

Given that $\alpha \beta = -\frac{k^2}{4}$, find the value of $k$.

The functions $f$ and $g$ are defined for $x \in \mathbb{R}$ by $f(x) = x - 2$ and $g(x) = ax + b$, where $a, b \in \mathbb{R}$.

Given that $f \circ g(2) = -3$ and $g \circ f(1) = 5$, find the value of $a$ and the value of $b$.

The functions $f$ and $g$ are defined for $x \in \mathbb{R}$ by $f(x) = 6x^2 - 12x + 1$ and $g(x) = -x + c$, where $c \in \mathbb{R}$.

This question asks you to explore properties of a family of curves of the type $y^2 = x^3 + ax + b$ for various values of $a$ and $b$, where $a,b \in \mathbb{N}$.

On the same set of axes, sketch the following curves for $-2 \leq x \leq 2$ and $-2 \leq y \leq 2$, clearly indicating any points of intersection with the coordinate axes.

Now, consider curves of the form $y^2 = x^3 + b$, for $x \geq -\sqrt[3]{b}$, where $b \in \mathbb{Z^+}$.

Next, consider the curve $y^2 = x^3 + x$, for $x \geq 0$.

The curve $y^2 = x^3 + x$ has two points of inflection. Due to the symmetry of the curve these points have the same $x$-coordinate.

$P(x,y)$ is defined to be a rational point on a curve if $x$ and $y$ are rational numbers. The tangent to the curve $y^2 = x^3 + ax + b$ at a rational point $P$ intersects the curve at another rational point $Q$. Let $C$ be the curve $y^2 = x^3 + 2$, for $x \geq -\sqrt[3]{2}$. The rational point $P(-1,-1)$ lies on $C$.