Let $f(x)=x^{2}+2$ and g(x)=x-3 , for $x \in \mathbb{R}$ .
1. Find f(3) . =   
2. Find $(g \circ f)(3)$ . =   
3. Find $g^{-1}(x)$ . =  (代数式) 

  Let $f(x)=x^{3}$ and g(x)=2 x-1 , for $x \in \mathbb{R}$ .
1. Find $g^{-1}(x)$ . =  (代数式) 
2. Find $(g \circ f)(x)$ . =  (代数式) 
3. Solve the equation $(g \circ f)(x)=0$ .≈   

  Let f(x)=4 x-3 andp 9hu-fe;2 u.taf5pyl $g(x)=2$ x , for $x \in \mathbb{R}$.
1. Write down the value of g(5) . =   
2. Find $(f \circ g)(x)$ . =  (代数式) 
3. Find $(f \circ g)^{-1}(x)$ . =  (代数式) 

  Let $f(x)=2 x$,$ g(x)=4 x+6 $ and $ h(x)=(f \circ g)(x)$ , for $x \in \mathbb{R} $.
1. Find $h(x) $.  (代数式) 
2. Find $ h^{-1}(x)$ .  (代数式) 

  Let $f(x)=\sqrt{x-3}$ , for $x \geq 3$ .
1. Find $f^{-1}(x)$ .

Let $g(x)=x^{2}+1$ , for $x \in \mathbb{R}$ . =  (代数式) 
2. Find $(g \circ f)(28)$ . =   
3. Write down the range of:
1. $f^{-1}$ ;
2. g .

  Let $f(x)=x^{3}+1 $ and g(x)=x-2 , for $x \in \mathbb{R}$ .
1. Find $f(2)$ . =   
2. Find $ f^{-1}(x)$ . =  (代数式) 
3. Solve $(f \circ g)(x)=0 .$   

  1. Find $f^{-1}(3)$ .

Let g be a function such that g^{-1} exists for all real numbers. =   
2. Given that g(9)=4 , find $\left(f \circ g^{-1}\right)(4) $. =   

The following diagram shows the graph of y= jk cn*ya 63y3qfg+nqd- aa83af(x) , for $-4 \leq x \leq 5$.



1. Write down the value of:
1. f(1) ;
2. $f^{-1}(2) $.
2. Find the domain of $f^{-1}$ .
3. Sketch the graph of $y=f^{-1}(x)$ on the same grid above.

  Let $f(x)=(x+2)^{3}$ , for $x \in \mathbb{R}$ .
1. Find $f^{-1}(x)$ .

Let g be a function so that $(f \circ g)(x)=27 x^{6}$ . =  (代数式) 
2. Find g(x) . =  (代数式) 

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) .

  The equation of a linepo(x jby:.nuo*xk.masv *36p $ L_{1}$ is 2 x+3 y=-5 .
1. Find the gradient of $L_{1}$ .

A second line, $L_{2}$ , is perpendicular to $L_{1}$ .   
2. State the gradient of $L_{2}$ .

The point $\mathrm{P}(4,0)$ lies on $L_{2}$ .   
3. Find the equation of $L_{2}$ , giving your answer in the form a x+b y+d=0 , where a, b, d $\in \mathbb{Z}$ .

The point \mathrm{Q} is the intersection of L_{1} and L_{2} . y=  (代数式) 
4. Find the coordinates of Q . (a,b) a =    b =   

  Two functions, f and g, are defined in the followings 3 )t hqaos3qb20g;zg table.

1. Write down the value of $f(2)$ .   
2. Find the value of $(g \circ f)(2)$ .   
3. Find the value of $g^{-1}(5)$ .   

The following diagram shows the a.hgtxv.n /ls1kwtxv ./ /amk: ig )/dgraph of y=f(x) , for $-1 \leq x \leq 2 $.



1. Write down the value of:
1. $f(1)$ ;
2. $f^{-1}(-2)$ .
2. Find $(f \circ f)(1)$ .
3. Sketch the graph of y=f(-x) on the same grid above.

  1. Express $2 x^{2}-8 x+9$ in the form $ a(x+b)^{2}+c$ where a, b, c $\in \mathbb{Z}$ .  (代数式) 
2. Given that f(x)=x-2 and $(g \circ f)(x)=2 x^{2}-8 x+9$ , find g(x) .  (代数式) 

  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 .  (代数式) 

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



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. A ≈   

  The function f is defi2) sh: wkiqwc+ned by $f(x)=\sqrt[3]{2 x+1}$ , for $-14 \leq x \leq 13$ .
1. Write down the range of f . [a,b] a =    b =   
2. Find an expression for $f^{-1}$ .  (代数式) 
3. Write down the domain and range of $f^{-1}$ .doamin [a,b] a =    b =    range [a,b] a =    b =   

The fThe function f is defir). ntuuh4ve+ned by


1. Determine whether or not f is continuous at x=1 .

The graph of the function g is obtained by applying the following transformations to the graph of f :
a horizontal translation 2 units to the left, followed by
a reflection in the x -axis, followed by
a vertical stretch by a factor of 3 .
2. Find g(x) .

  1. Express 3 x^{2}+18 x+20 in the form $a(x+b)^{2}+c$ where a, b, c $\in \mathbb{Z}$ .  (代数式) 
2. Given that f(x)=x+3 and ($g \circ f$)(x)=3 x^{2}+18 x+20 , find g(x) .  (代数式) 

  The functions f and g are defined such that u f5s1kfl4hn1$ f(x)=\frac{x-2}{3}$ and $g(x)=12 x+4 $.
1. Show that $(g \circ f)(x)=4 x-4$.  (代数式) 
2. Given that $(g \circ f)^{-1}(a)=10$ , find the value of a .   

Let $f(x)=\frac{1}{4} x^{2}-2 and g(x)=x^{2}-4 , for $x \in \mathbb{R}$ .
1. Show that $(f \circ g)(x)=\frac{1}{4} x^{4}-2 x^{2}+2$ . __
2. On the following grid, sketch the graph of $y=(f \circ g)(x) $, for $0 \leq x \leq 3 $. __

Let $f(x)=\frac{1}{4}$ $x^{2}-2 $ and g(x)=x^{2}-4 , for $x \in \mathbb{R}$ .
1. Show that $(f \circ g)(x)=\frac{1}{4} x^{4}-2 x^{2}+2$ .
2. On the following grid, sketch the graph of $y=(f \circ g)(x)$ , for $ 0 \leq x \leq 3 $.

The area, A , of a given square can be represented by the funl7y b.dwwsl02 ction

$A(P)=\frac{P^{2}}{16}$, $\quad P \geq 0$,

where P is the perimeter of the square.
The graph of the function A , for $0 \leq P \leq 20$ , is shown below.

1. Find the value of A(20) .
2. On the grid above, draw the graph of the inverse function, $A^{-1}$ .
3. In the context of the question, explain the meaning of $A^{-1}(4)=8$ .

The circumference, C , of a given circle can be represented by the f3je(00-y jpusdly5s zunction

$C(A)=2 \pi \sqrt{\frac{A}{\pi}}, \quad A \geq 0$

where A is the area of the circle.
The graph of the function C , for $0 \leq A \leq 24 $, is shown below.

1. Find the value of C(24) .
2. On the grid above, draw the graph of the inverse function, $C^{-1}$ .
3. In the context of the question, explain the meaning of $C^{-1}(2 \pi)=\pi$ .

The ocean pressure, P , under sea levijj;, t(+tu,p ve99ifd ) lz/0rzmukeel can be modelled by the function

$P(D)=\frac{D}{10}+1$

where D is the distance in metres below sea level and P is measured in atmospheres.
A submarine cruising near the surface is submerged according to the function

D(t)=10+5 t

where t is measured in minutes and D is the distance the submarine is below sea level, measured in metres.
1. Find the composite function $P \circ D$ and explain what it means in the context of this question
2. Find and interpret $(P \circ D)(10)$ .

A tyre manufacturing company hx6 2:obyapb2bqiqn+8 as found that the number of tyres it produces, N , can be modqbp +a ynq:2o2xi6b8belled by the function

N(t)=3 t-9

where t is the number of hours the factory operates per day, with a minimum of 3 hours.
The profit the company makes, P , in dollars, depends on the number of tyres produced, and is modelled by the function

P(N)=60 N-850

where N is the number of tyres produced.
1. Find the company's profit or loss if it operates for 6 hours per day.
2. Find the company's profit in terms of the hours of operation per day, t .
3. Determine the number of hours the company needs to operate the factory per day in order to earn a positive profit. Give your answer to the nearest hour.

A town is planning to construho*k/u10 i6h(vgs)bgbj f;usct a jogging path in a grass field 170 $\mathrm{~m} $ long and 70$\mathrm{~m}$ wide. The path is to be the shape of a rectangle with two semicircles of radius x , as shown in the diagram. The sides of the rectangle connecting the circles are to be 100$ \mathrm{~m}$ long.



1. Write down a function, P , (in metres) for the perimeter of the jogging path, in terms of the radius, x .
2. Determine the domain and range of P , taking into consideration the dimensions of the grass field.
3. Find an equation for the inverse function $P^{-1}(x)$ . Express your answer in the form $ P^{-1}(x)=m x+c$ .

The designers of the path are deciding whether the total length of the path should be 300 $\mathrm{~m}$, 400 $\mathrm{~m}$ , or 500 $\mathrm{~m}$ . The designers want to maximise the perimeter of the path, but fit the path in the grass field.
4. Determine which length is most suitable, given the dimensions of the grass field.

Harry is planning on constructing a glass windzaz)ty hw xob7 7 63cnu+thwe6:qcw10ow in one of the outer walls of his house. The dimensions of the wall space available are 2m x 2m. Harry wants the window to be in the shape shown in the diagram below. The 7whn1oxa w7tw)zc03zh:qybeu 6+c t6 bottom section is a rectangle and the top part is a semicircle of radius x m. Harry wants the height of the rectangle to be fixed at 1 m.



1. Write down a function P (in metres) for the perimeter of the window in terms of the radius, x .
2. Determine the domain and range of P , taking into consideration the dimensions of the available wall.
3. Find an equation for the inverse function $P^{-1}(x)$ .

Harry wants to maximise the size of the window, however the window frame that covers the perimeter of the window can only be 5,6 , or 7 metres long, due to manufacturing restrictions.
4. Determine which perimeter length is the best option for Harry.

The function f is defined v-ehj+ bj7y )zas $f(x)=\sqrt{\frac{6+2 x}{6-2 x}}$ , for $-3 \leq x<3 $. Find the inverse function of f , stating its domain and range.

Let f(x)=$\frac{5}{x+1}$ , for $ x \neq-1 $.
1. For the graph of f , find:
1. the x -intercept;
2. the y -intercept;
3. the equation of the vertical asymptote.

Let g(x)=x-3 , for x $\in \mathbb{R}$. The graphs of f and g intersect at points $\mathrm{A} $ and $ \mathrm{B}$ .
2. Find the coordinates of A and B .
3. Find the equation of the straight line that passes through A and B , giving your answer in the form y=m x+c .
4. Write down the gradient of the line that is perpendicular to the line passing through A and B .

The following diagram shows the graph of y=f(x). Theahtxsha2 1m ;-z+mg2phf*v2l graph has a horizontal asymptoms 2* a1v2 g+2lpax;hh-fthzmte at y=-2. The graph crosses the x-axis at x=-2 and x=2 , and the y -axis at y=2 .


$\text { On the following set of axes, sketch the graph of } y=[f(x)]^{2}-1 \text {, clearly showing any asymptotes with their equations and the coordinates of any local maxima or minima. }$

1. Write down the domaint 8i a0dopm29( jbv7)w5mtlpu vzzr)- and range of the logarithmic 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 .

Consider the graphs of1c*gqn)tg5 h gqeg5 h- $y=-\frac{1}{2}|x|$ 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 functiom6 agyy k.s.j.i) uqk*n $f(x)=x^{2} \arcsin (x)$ , 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 $\left|x^{2} \arcsin (x)\right|>0.5$ .

$\text { The following diagram shows the graph of } y=f(x) \text {. The graph has a horizontal asymptote at } y=-2 \text {. The graph crosses the } x \text {-axis at } x=-1 \text { and } x=1 \text {, and the } y \text {-axis at } y=2 \text {. }$


$\text { On the following set of axes, sketch the graph of } y=[f(x)]^{2}-2 \text {, clearly showing any asymptotes with their equations and the coordinates of any local maxima or minima. }$

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 $\sum_{k=1}^{n} x_{k}=992$ .

  $\text { Let } f(x)=a x^{3}+b x^{2}+c x+d \text {, for } x \in \mathbb{R} \text {, where } a, b, c, d \in \mathbb{Q} \text {. The diagram below shows part of the graph of } y=f(x) \text {. }$



1. Using the information shown in the diagram, find the values of a, b, c and d . a =    b =    c =   
2. Let $g(x)=-\frac{3}{4} f(-2 x+1)$ .
1. Find the coordinates of the points where the graph of y=g(x) intercepts the x -axis. x =         
2. Find the y -intercept of the graph of y=g(x) .P (A,B) A =    B =   

Let $ f(x)=x^{4}-x^{3}-5 x^{2}+3 x+2$ , for $x \in \mathbb{R}$ .
1. Solve the inequality f(x)<0 .
2. For the graph of y=f(x) , find the coordinates of the local maximum point. Round your answers to three significant figures.

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 the equation $f^{-1}(x)=-1$ .

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.

The function f is defz/10 lwdzda1sined 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 75- ovkt p c *,oy9l z)h66,ormuoyjhpv2pk .x$ 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 $x+4=\frac{5}{|x-2|}$ .

It is given that $f(x)=2 x^{4}+5 x^{3}+a x^{2}+b x+4$ , for x $\in \mathbb{R}$ , where a, b $\in \mathbb{Z}^{+}$ .
1. Given that $x^{2}+x-2$ is a factor of f(x) , find the values of a and b .
2. Factorise f(x) into a product of linear factors.
3. Sketch the graph of y=f(x) , labeling the maximum and minimum points and the x and y intercepts.
4. Using your graph, state the range of values of c for which f(x)=c has exactly four distinct real roots.

  The function f is defineyuc6x mr 37zsb /2auz.gv-(egd 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.  (数值) 
2. For this value of a , find an expression for $f^{-1}(x)$  (代数式)  stating its domain

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 $\left(f^{-1} \circ g\right)(x)<0.5$ .

A function f(x) is defineq nwaa1u*f +7m npuv63w:ng*rd by $f(x)=\arccos \left(\frac{x^{2}-1}{x^{2}+1}\right)$, $x \in \mathbb{R}$
1. Show that f is an even function.
2. Find the equation of the horizontal asymptote to the graph of y=f(x) .
3. 1. Show that $f^{\prime}(x)=-\frac{2 x}{\sqrt{x^{2}}\left(x^{2}+1\right)} $ for $x \in \mathbb{R}$, $x \neq 0$ .
2. Using the expression for $f^{\prime}(x) $ and the result $ \sqrt{x^{2}}=|x|$ , show that f is increasing for x<0 .

A function g is defined by $g(x)=\arccos \left(\frac{x^{2}-1}{x^{2}+1}\right)$, $x \in \mathbb{R}$, $x \geq 0$ .
4. Find the range of g .
5. Find an expression for $g^{-1}(x)$ .
6. State the domain of $ g^{-1}(x)$ .
7. Sketch the graph of $y=g^{-1}(x) $. Clearly indicating any asymptotes with their equations and stating the values of any axes intercepts.