help

**tamilmassive** · Apr 2nd 2008, 04:14 PM

I am totally clueless abt these problems..any help would be appreciated.Thank you. I have the two problems attached.

**Soroban** · Apr 2nd 2008, 06:40 PM

Hello, tamilmassive!

3. Let $f(x) \:=\:\frac{2-x}{3-2x}\quad\left(x \neq \frac{3}{2}\right)$

Find a function $g(x)$ of the form . $\frac{ax +b}{cx+d}$ .so that: $g(g(x)) \:=\:f(x).$
Specify at what point $g$ is defined.

I will assume that $a,b,c,d$ are integers.

We have: . $g(g(x)) \;=\;\frac{a\left(\dfrac{ax+b}{cx+d}\right) + b}{c\left(\dfrac{ax+b}{cx+d}\right) + d}$

Multiply top and bottom by $(cx+d)\!:\;\;\frac{a(ax+b) + b(cx+d)}{c(ax+b) + d(cx+d)} \;=\;\frac{(a^2+bc)x + (ab+bd)}{(ac + cd)x + (bc+d^2)}$

Since this is to be equal to $f(x)$ , we have: . $\frac{(a^2+bc)x + b(a+d)}{c(a+d)x + (bc+d^2)} \;=\;\frac{-x+2}{-2x+3}$

Equating coefficients: . $\begin{array}{ccccccccc}<br /> {\color{blue}(1)} & a^2+bc &=& -1 & & {\color{blue}(2)} & b(a+d) &=& 2 \\<br /> {\color{blue}(3)} & c(a+d) &=&-2 & & {\color{blue}(4)} & bc+d^2 &=& 3 \end{array}$

Divide (3) by (2): . $\frac{c(a+d)}{b(a+d)} \:=\:\frac{\text{-}2}{2}\quad\Rightarrow\quad \frac{c}{b} \:=\:-1\quad\Rightarrow\quad c \:=\:-b$

Substitute into (1): . $a^2-b^2 \:=\:-1$

. . Let $a \,= \,0,\;b \,= \,1$ . . . then $c \,= \,-1$

Substitute into (4): . $-1 + d^2\:=\:3\quad\Rightarrow\quad d \,=\,2$

Therefore: . $\boxed{g(x) \;=\;\frac{1}{2-x}}$

$g(x)$ is defined for all real $x \neq 2.$

**tamilmassive** · Apr 3rd 2008, 03:31 AM

thank you for your help

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