1. Sketch the graph of : ubrohw/(vu14 2j vpsivdy:6 $y=(x-2{)}^2-4|x-2|-1$ , for $-3\le x\le 7$ .
2. Hence, or otherwise, solve the equation $(x-2{)}^2-4|x-2|-1=0$ .

Consider the functiojv)jz pb0wilf- 9-9m-r z m6ogn $f(x)=\frac{e^x-e^{-x}}{2}$, $x\in \mathbb{R}$.
1. Show that f is an odd function.

Now, consider the function g given by $g(x)=\frac{x^4+2}{2x}$, $x\in \mathbb{R}$, $x\ne 0$ .
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, :frkw4-dcplh *xt3-e P , can be modelled by the equation

$P_t=P_0{e}^{kt}$

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) r vvts3:a.jn $y=-\frac{1}{2}|x|$ 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 8+c kigtqp3 htjsx 365$f(x)=x^2\arcsin (x)$ , for $-1\le x\le 1$ .
1. Sketch the graph of y=f(x) .
2. Write down the range of f .
3. Solve the inequality $|x^2\arcsin (x)|>0.5$ .

Let $f(x)=\frac{9-12x}{cx-20}$ , for x$\ne \frac{20}{c}$ , where c$\ne 0$ .
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 d zhzqy2,w 45 4:hiawklefined by $f(x)=1+\frac{4x}{x+3}$ , for $x\ne -3$.
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 ) -;jygq4ubht:pam3rfe3,f g$y=-\left|\frac{5}{x-2}\right|$ 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 inequality0vstfr p -7hi0 $x^2\ge 2x+3$.
2. Use mathematical induction to prove that $2^n>n^2-2$ for all $n\in {\mathbb{Z}}^{+}$,$n\ge 3$ .

Let $f(x)=x^4-0.4x^3-2.85x^2+0.9x+1.35$, 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 $g(x)=\frac{2}{3}\sin (2x-1)+\frac{1}{2}$, $\frac{1}{2}-\frac{\pi }{4}\le x\le \frac{1}{2}+\frac{\pi }{4}$.
4. 1. Find an expression for g^{-1} and state its domain.
2. Solve $(f^{-1}\circ g)(x)<0.5$ .

Consider the function mjeh:c6xg8m ;.u s0 hqdefined by $f(x)=(2x-6)\ln (x+3)+x$ 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 .