1. Sketch the graph osovoa)0 6yar(f $y=(x-2)^{2}-4|x-2|-1$ , for $-3 \leq x \leq 7$ .
2. Hence, or otherwise, solve the equation $(x-2)^{2}-4|x-2|-1=0$ .

Consider the function k thb7*tomj gu3w52jr d8z(4uxnp m11 $f(x)=\frac{e^{x}-e^{-x}}{2}$, $x \in \mathbb{R} $.
1. Show that f is an odd function.

Now, consider the function g given by $g(x)=\frac{x^{4}+2}{2 x}$, $x \in \mathbb{R}$, $x \neq 0$ .
2. By considering the graph of y=f(x)-g(x) , solve f(x)>g(x) for $x \in \mathbb{R} $.

  A population of endangered snow leopards, P , can be modelled by the equationpczpy we mk* fj.gx6q14w+x (-

$P_{t}=P_{0} e^{k t}$

where $P_{0}$ is the initial population, and t is measured in years.
1. After one year, it is estimated that $\frac{P_{1}}{P_{0}}=0.93 $.
1. Find the value of k .
2. Interpret the meaning of the value of k .≈   
2. Find the least number of whole years for which $\frac{P_{t}}{P_{0}}<0.50$ . t =   

Consider the graphs of z ei /x-x0lfa/ $y=-\frac{1}{2}|x|$ and y=2|x|-a , where $a \in \mathbb{Z}^{+}$ .
1. Sketch the graphs on the same set of axes.
2. Given that the graphs enclose a region of area 40 square units, find the value of a .

Consider the function 1n7; now3viui $ f(x)=x^{2} \arcsin (x)$ , for $ -1 \leq x \leq 1$ .
1. Sketch the graph of y=f(x) .
2. Write down the range of f .
3. Solve the inequality $\left|x^{2} \arcsin (x)\right|>0.5$ .

Let $f(x)=\frac{9-12 x}{c x-20}$ , for x$ \neq \frac{20}{c}$ , where c$\neq 0$ .
1. The line x=5 is a vertical asymptote to the graph of y=f(x) .
1. Find the value of c .
2. Write down the equation of the horizontal asymptote to the graph of y=f(x) .
2. The line y=h , where h $\in \mathbb{R}$ , intersects the graph of y=|f(x)| at exactly one point. Find the possible values of h .

The function f is de* v: e fxmxg8x,:ch1elfined by $f(x)=1+\frac{4 x}{x+3}$ , for $x \neq-3 $.
1. Sketch the graph of y=f(x) , indicating clearly any asymptotes and points of intersection with the x and y axes.
2. Find an expression for $f^{-1}$ .
3. Find all values of x for which $f(x)=f^{-1}(x)$ .
4. Solve the inequality $|f(x)|<2 $.
5. Solve the inequality $f(|x|)<2 $.

1. Sketch the curve xs wjm675t 8mz $y=-\left|\frac{5}{x-2}\right|$ and line y=-x-4 on the same axes, clearly indicating any x and y intercepts and any asymptotes.
2. Find the exact solutions to the equation $x+4=\frac{5}{|x-2|}$ .

1. Solve the inequali.hz 9g4oih-;d 6qwvz jsy3 md-ty $x^{2} \geq 2 x+3 $.
2. Use mathematical induction to prove that $2^{n}>n^{2}-2$ for all $n \in \mathbb{Z}^{+}$,$ n \geq 3$ .

Let $f(x)=x^{4}-0.4 x^{3}-2.85 x^{2}+0.9 x+1.35$, for x $\in \mathbb{R} $.
1. Find the solutions for f(x)>0 .
2. For the graph of y=f(x) ,
1. find the coordinates of local minimum and maximum points.
2. find the x -coordinates of the points of inflexion.

The domain of f is now restricted to [a, b] where a, b $\in \mathbb{R}^{+}$ .
3. 1. Write down the smallest value of a>0 and the largest value of b>0 for which f has an inverse. Give your answers correct to three significant figures.
2. For these values of a and b , sketch the graphs of y=f(x) and y=$f^{-1}(x)$ on the same set of axes, showing clearly the coordinates of the end points of each curve.
3. Solve f^{-1}(x)=0.5 .

Let $g(x)=\frac{2}{3} \sin (2 x-1)+\frac{1}{2}$, $\frac{1}{2}-\frac{\pi}{4} \leq x \leq \frac{1}{2}+\frac{\pi}{4} $.
4. 1. Find an expression for g^{-1} and state its domain.
2. Solve $ \left(f^{-1} \circ g\right)(x)<0.5$ .

Consider the function defined 6jh 5pg d-+ahfby $f(x)=(2 x-6) \ln (x+3)+x $ for $ x \in \mathbb{R}$, x>p .
1. Find the value of p .
2. Find an expression for $f^{\prime}(x)$ .

The graph of y=f(x) has no points of inflexion.
3. Determine if the graph of y=f(x) is concave down or concave up over its domain.

The function g is defined by $g(x)=3 \ln \left(\frac{1}{x+3}\right)+x $, for $ x \in \mathbb{R}$, x>-3 .
4. Find an expression for $g^{\prime}(x)$ .
5. Find the horizontal and vertical asymptotes of $g^{\prime}(x) $.
6. Find the exact value of the minimum of y=g(x) .
7. Solve f(x)-3 .