Let $f(x)=0.2 e^{x+2}-4$ , for $-3 \leq x \leq 2 $.
1. On the following grid, sketch the graph of y=f(x) .

2. Find the coordinates of:
1. the x -intercept;
2. the y -intercept.

The graph of f is reflected in the x -axis and then translated by the vector \binom{1}{2} to obtain the graph of a function g .
3. Find g(x) .

  1. Write the expressi t/7e-;cs jevqon $4 \ln 2-\ln 8$ in the form of $\ln k$ , where $k \in \mathbb{Z}$.   
2. Hence, or otherwise, solve $4 \ln 2-\ln 8=-\ln (2 x)$ .   

  $\text { Solve the equation } \log _{3} x-\log _{3} 5=1+\log _{3} 4 \text { for } x \text {. }$   

  $\text { Solve the equation } \log _{5} x-\log _{5} 4=2+\log _{5} 3 \text { for } x \text {. }$   

  Let $f(x)=e^{x}-2$ and $ g(x)=3 x+k$ , for $x \in \mathbb{R}$ , where k is a constant.
1. Find $(g \circ f)(x)$ .  (代数式) 
2. Given that $\lim _{x \rightarrow-\infty}(g \circ f)(x)=-4$ , find the value of k .   

  It has been suggested that the number of visits (the number of hits) to v.hu5s nix7 y-;g7btpd*o ) ksa newly lau)*y-go dk7invhtp s.bs; x57unched website can be modelled by an equation of the form H=a N^{b} , where H is the number of hits in each month, N is the month number, a and b are constants.
To test this model, a new website was created and the number of hits in each of the first 10 months were recorded.
The results are shown in the following table for N=1 and N=2 .

1. 1. Write down the value of a .   
2. Find the value of b , giving your answer to five significant figures.≈   
2. Use this model to estimate the number of hits in the 8th month. Give your answer to the nearest integer.≈   

  $\text { Let } f(x)=\frac{\ln (x+2)}{2} \text {, for } x>-2 \text {. The diagram below shows part of the graph of } f \text {. }$



1. Find the coordinates of:
1. the x -intercept; (a,b) a =    b =   
2. the y -intercept.(a,b) a =    b =   
2. Find the equation of the vertical asymptote to the graph of f . x =   
3. Find the area of the region enclosed by the graph of f , the x -axis and the y -axis.≈   

  Atmospheric pressure, P , in kPa, decreases exponentially with inc xj;kmk:x9e lj75xz /creasing height abejc kk/lm x5z:9;jxx 7ove sea level, h . The atmospheric pressure can be modelled by the function

P(h)=101 $\times\left(\frac{25}{22}\right)^{-h}$,

where h is the height above sea level in kilometres.
1. Write down the exact atmospheric pressure at sea level, in $\mathrm{kPa}$ .

Mount Kosciuszko is the highest mountain in Australia with a height of 2228 metres above sea level at the top.    kPa
2. Calculate the atmospheric pressure at the top of the Mount Kosciuszko.≈   
3. Calculate the height where the atmospheric pressure is equal to 10 $\mathrm{kPa}$ .   

  A loaf pan is made in the shape of a cyliny.qdm0 s oadpld,;sc+ y8: g,uder. The pan has a diameter of 24 cm and a+ c.u,sdopag;m:08,qs ldyy d height of 5 m.



1. Calculate the volume of this pan.

Gloria prepares enough bread dough to exactly fill the pan. The dough was in the shape of a sphere. ≈   
2. Find the radius of the sphere in $\mathrm{cm}$ , correct to one decimal place.

The bread was cooked in a hot oven. Once taken out of the oven, the bread was left in the kitchen. The temperature, T , of the bread, in degrees Celsius, ${ }^{\circ} \mathrm{C}$ , can be modelled by the function

$T(t)=a \times(1.51)^{-\frac{t}{3}}+21, \quad t \geq 0$ ≈   

where a is a constant and t is the time, in minutes, since the bread was taken out of the oven. When the bread was taken out of the oven its temperature was 205^{\circ} \mathrm{C} .
3. Find the value of a .   
4. Find the temperature that the bread will be 10 minutes after it is taken out of the oven.

The bread can be eaten once its temperature drops to $35^{\circ} \mathrm{C}$ .
5. Calculate, to the nearest minute, the time since the bread was taken out of the oven until it can be eaten. ≈   
6. In the context of this model, state what the value of 21 represents. ≈   

  A population of goldfish decreases exponentially. The size1; pcs ebzhl/se ;+.ht/tl2 kl of the population, P , after t days is modelled by thbs/ph+ kthet2;l 1zs; .c/l ele function

$P(t)=8000 \times 2^{-t}+100, \quad t \geq 0 \text {. }$

1. Write down the exact size of the initial population.   
2. Find the size of the population after 5 days.   
3. Calculate the time it will take for the size of the population to decrease to 120 .

The population will stabilize when it reaches a size of n .   
4. Write down the value of n .   

  A population of 50 hamsters wo 3 m(48iexgdfas introduced to a new town. One month later, the number of hamsters was 62 . The number mfx4i3d og(8e of hamsters, P , can be modelled by the function

P(t)=50 $\times b^{t}, \quad t \geq 0$

where t is the time, in months, since the hamsters were introduced to the town.
1. Find the value of b .   
2. Calculate the number of hamsters in the town after 6 months.

A wildlife specialist estimates that the town has enough water and food to support a maximum population of 2000 hamsters. ≈   
3. Calculate the number of months it takes for the hamster population to reach this maximum. ≈   

  $\text { Find the values of } x \text { when } 27^{x+2}=\left(\frac{1}{9}\right)^{2 x+4} \text {. }$   

  $\text { Solve } \log _{6}(x)+\log _{6}(x-5)=2 \text {, for } x>5$   

  $\text { Solve the equation } \log _{2}\left(x^{2}-2 x+1\right)=1+\log _{2}(x-1) \text {. }$   

  $\text { Solve the equation } \log _{3}\left(x^{2}-4 x+4\right)=1+\log _{3}(x-2) \text {. }$   

  The number of French words, z d( utsw0smtx0546v yN , that are remembered by students after the completion of t0smz5ywd(sv t6 04ux a French language course decreases exponentially over time. This data can be modelled by the function

N(t)=a $\times b^{-t}+450$,

where a and b are positive constants, and t is the time in years since a student completed the French language course.
Immediately after completion, a student remembers 4200 French words.
1. Find the value of a .

After 4 years a student remembers only 1600 French words.   
2. Find the value of b .

The number of French words a student remembers never decreases below a certain number of words, n . ≈   
3. Write down the value of n .   

  The number of users on a new social network on January 1st 2w0:bci8nny)n+ qbc h: y qv+*x018 was 320000 . One year later, on January 1 st 2019 , the number of users on this network is estimated to be 350000 . The number of the usei* y :b0 :n)bnvynwc q+xhqc+8rs on this network, N , can be modelled by the function

$N(t)=320000 \times b^{t}, \quad t \geq 0$,

where t is the number of years since January 1st 2018 and b is a constant.
1. Find the exact value of b .   
2. Estimate the number of the users on this network there will be in 2023 .   
3. Determine the year during which the number of the social network users reaches one million.   

  $\text { Find the values of } x \text { when } 25^{x^{2}-2 x}=\left(\frac{1}{125}\right)^{4 x+2} \text {. }$      

  1 $\text { Solve the equation } 9^{x}+2 \cdot 3^{x+1}=1$
The solution is $x$ =  (数值) 

Consider the functioqv dfgcu.e+b tv/sldn-w ; 6;dgp;46en $f(x)=\frac{e^{x}-e^{-x}}{2}$,$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} $.

1. Write down the domain and range of the logarithe8tdqas9 941e/(yjpu,geoa v mic function $y=\log _{a}$ x where a>0 and $a \neq 1$ .
2. Given that $\log _{x^{2}} y=9 \log _{y}\left(x^{2}\right)$ , find all the possible expressions of y as a function of x .

  A population of endangered snow leopards, P fhowkp*lb sq3,kby;pn0m (zw. gl/l22b+ 7fm, can be modelled by the equation

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

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 $\frac{P_{t}}{P_{0}}<0.50$ .   

  Consider $ f(x)=\log _{k}\left(8 x-2 x^{2}\right)$ , for 00 .
The equation f(x)=3 has exactly one solution. Find the value of k .   

  The following table shows values of lnx and lb*o *x;j(u- wrv;zf dyny.



The relationship between ln x and ln y can be modelled by the regression equation ln y=a ln x+b .
1. Find the value of a and the value of b .a ≈    b ≈   
2. Use the regression equation to estimate the value of y when x=4.12 .

The relationship between x and y can be modelled using the formula $y=p x^{q}$ , where $p \neq 0$ and $q \neq 0$, $q \neq 1$ . ≈   
3. Expressing ln y in terms of ln x , find the value of p and the value of q . p ≈    q ≈   

  A study records the number of chickens and ducks on z5ri(c6q6bkca farm after t years, starting on 1 Januakz qr6 6icbc5(ry, 1997.Let c be the number of chickens on the farm after t years. The following table shows the number of chickens after t years.




The relationship between the variables can be modelled by the regression equation c=a t+b , where a and b are constants.
1. Find the value of a and the value of b . a ≈    b≈   
2. Use the regression equation to estimate the number of chickens on the farm when t=6 .

Let d be the number of ducks on the farm after t years. The number of ducks can be modelled by the equation $d=400 e^{-k t}$ , where k is a constant.≈   
3. Find the number of ducks on the farm on 1 January, 1997.   
4. After six years, there are 377 ducks on the farm. Find the value of k .≈   
5. Find the year during which the number of chickens and ducks were the same.   

  $\text { Solve the equation } 15^{4 a}=81^{a+2} \text { for } a \text {. Express your answer in terms of } \ln 3 \text { and } \ln 5 \text {. }$  (数值) 

  $\text { Solve the equation } 14^{6 x}=64^{x+3} \text { for } x \text {. Express your answer in terms of } \ln 2 \text { and } \ln 7 \text {. }$  (数值) 

  Let $f(x)=e^{3 \sin \left(\frac{\pi x}{4}\right)}$ , for x>0 .
The k th maximum point on the graph of f has x -coordinate $x_{k}$ , where $ k \in \mathbb{Z}^{+}$ .
1. Given that $x_{k+1}=x_{k}+d$ , find d .   
2. Hence find the value of n such that $\sum_{k=1}^{n} x_{k}=992$ .   

Consider $ f(x)=\frac{1}{2}-\ln \left(\sqrt{x^{2}-4}\right) $.
1. Find the largest possible domain D for f to be a function.

The function f is defined by $f(x)=\frac{1}{2}-\ln \left(\sqrt{x^{2}-4}\right)$, for $x \in D $.
2. Sketch the graph of y=f(x) , showing clearly the equations of asymptotes and the coordinates of any intercepts with the axes.
3. Explain why f is an even function.
4. Explain why the inverse function $f^{-1}$ does not exist.

The function g is defined by $g(x)=\frac{1}{2}-\ln \left(\sqrt{x^{2}-4}\right)$ , for $x \in(2, \infty)$ .
5. Find the inverse function $g^{-1}$ and state its domain.

  1. Show that $3 \log _{a^{3}} x=\log _{a} x$ where a, $x \in \mathbb{R}^{+}$ .

2. It is given that $\log _{2} y+\log _{8} 4 x^{2}+\log _{8} 2 x=0$ .
Express y in terms of x . Give your answer in the form $y=b x^{c}$ where b, c are constants. $y$ =  (代数式) 

3. The region R , is bounded by the graph of the function found in part (b), the x -axis, and the lines x=1 and x=k where k>1 . The area of R is $ \frac{3}{2}$ . Find the value of $k$ =  (数值) 

The function f is 2w5qlko/ iph p8:qfhfw8 5jtru. jm9* 3ik h)pdefined by $f(x)=e^{2 x}-4 e^{x}+2$ , for $ x \in \mathbb{R}$, $x \leq a$ , where $a \in \mathbb{R}$ .
Part of the graph of y=f(x) is shown in the following diagram.

1. Find the largest value of a such that f has an inverse function.
2. For this value of a , find an expression for $ f^{-1}(x)$ , stating its domain.

Consider the function defined **ijj eu k4rx1(v8 +exdqd. zcby $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$ .