The function f is o 2l ribf6.) r h8:jcic,0mfzcmf the form $f(x)=\frac{a x+b}{2 x+c}$ , for $x \neq-\frac{c}{2}$ , where a, b, c $\in \mathbb{Z}$ . Given that the graph of y=f(x) has asymptotes x=-5 and y=2 , and that the point $\mathrm{P}\left(1,-\frac{1}{12}\right)$ lies on the graph, find the values of a, b and c . a =    b =    c =   

  A rational function is defined bfkqdg ;i7d/ 9rwkvn jjg(873hy $ f(x)=a+\frac{b}{x-c}$ , for $x \neq c$ , where a, b, c $\in \mathbb{Z}$. The following diagram represents the graph of y=f(x) .



Using the information on the graph,
1. state the value of a and the value of c ; a =    c =   
2. find the value of b . b =   

Let $f(x)=\frac{3 x-6}{x+1}$ , for $ x \neq-1$ .
1. For the graph of y=f(x) , find the coordinates of:
1. the x -intercept;
2. the y -intercept.
2. For the graph of y=f(x) , find the equation of:
1. the horizontal asymptote;
2. the vertical asymptote.
3. Hence sketch the graph of y=f(x) .

  Let $f(x)=\frac{2}{x-a}+b$ , for $x \neq a$ .
The line x=1 is a vertical asymptote of the graph of y=f(x) .
1. Write down the value of a .

The graph of y=f(x) passes through the point $\mathrm{P}(0,3)$ .   
2. Find the value of b .   
3. Find $\lim _{x \rightarrow \infty} f(x)$ .   

The function f is gpf ohn5vp9l 0xc0oea,07 f:yxiven by $f(x)=\frac{4 x^{2}-128}{x^{2}-16}$ , for $x \neq \pm 4$ .
1. Prove that f is an even function.
2. 1. Sketch the graph of y=f(x) .
2. Write down the range of f .

  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; p(a,b) a =    b =   
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 . A (a,b) a =    b =    B(c,d) c =    d =   
3. Find the equation of the straight line that passes through A and B , giving your answer in the form y=m x+c . y =  (代数式) 
4. Write down the gradient of the line that is perpendicular to the line passing through A and B .   

Let $ f(x)=\frac{3 x+4}{2 x-5}$ , for $x \neq \frac{5}{2} $.
1. For the graph of y=f(x) , find the coordinates of:
1. the x -intercept;
2. the y -intercept.
2. For the graph of y=f(x) , find the equation of:
1. the horizontal asymptote;
2. the vertical asymptote.
3. Hence sketch the graph of y=f(x) .
4. Find $ f^{-1}(x) $.

The graphs of f(x) and $f^{-1}(x)$ have two points of intersection.
5. Find the coordinates of the two points of intersection. Give your answers to two decimal places.
6. Hence find the equation of the perpendicular bisector to the two points of intersection.

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 defk:b (y2g5.yfr -3v z ohpknww92fgr8bined 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 d1 31hbgj-;lo1m pv0hq o m wad.g2t(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|}$ .

Consider the function detgk /ts6-mf 8jcc vg2 e,vq-+afined by $f(x)=\frac{7}{x^{2}+6 x-7}$ for $x \in \mathbb{R}$, x $\neq-7, x \neq 1$ .
1. Sketch the graph of y=f(x) , showing the values of any axes intercepts, the coordinates of any local maxima and minima, and the graphs of any asymptotes.

Next, consider the function g defined by $g(x)=\frac{7}{x^{2}+6 x-7} $ for $x \in \mathbb{R}$, x>1 .
2. Show that $g^{-1}(x)=-3+\sqrt{\frac{16 x+7}{x}}$ .
3. State the domain of g^{-1} .

Now, consider the function h defined by $h(x)=\arccos \left(\frac{x}{7}\right)$.
4. Given that $(h \circ g)(a)=\frac{\pi}{3}$ , find the value of a . Give your answer in the form $p+q \sqrt{2}$ where p, q $\in \mathbb{Z}$ .

Consider the function defined byd/1+f)abejj*3 e+bi 0r f yzqf $f(x)=(2 x-6) \ln (x+3)+x$ 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.