# Thread: For which numbers a,b,c,d will the function f(x) satisfy...

1. ## For which numbers a,b,c,d will the function f(x) satisfy...

Problem:
For which numbers $\displaystyle a,b,c,d$ will the function
$\displaystyle f(x)=\frac{ax+b}{cx+d}$
satisfy $\displaystyle f(f(x))=x$ for all $\displaystyle x$?

------------------------------------------------------------------------
Attempt:
$\displaystyle f(f(x))=\frac{a\left[\frac{ax+b}{cx+d}\right]+b}{c\left[\frac{ax+b}{cx+d}\right]+d}\\ = \frac{\left[\frac{a^2x+ab}{cx+d}\right]+b}{\left[\frac{acx+bc}{cx+d}\right]+d}\\ =\frac{(a^2+bc)x+(ab+bd)}{(ac+cd)x+(bc+d^2)}$

By using polynomial division I get the following result:

$\displaystyle \left(\frac{a^2+bc}{ac+cd}\right)+\frac{(ab+bd)-(bc+d^2)\left(\frac{a^2+bc}{ac+cd}\right)}{(ac+cd) x+(bc+d^2)}$

By doing some hopefully correct algebra I get:

$\displaystyle (a^2+bc)x+(ab+bd)=x$
1. $\displaystyle (ab+bd)=0$
So, $\displaystyle a=-d$

2. $\displaystyle (a^2+bc)=1$
So, $\displaystyle a=\sqrt{1-bc}$

If $\displaystyle b$ and $\displaystyle c$ have opposite signs, their product is negative, and they can be any number.

If on the other hand, their signs are the same, their product is positive and:
$\displaystyle -1 \leq b \leq 1$
$\displaystyle -1 \leq c \leq 1$
-----------------------------------------------------------------------

Am I making this more complicated than it is?
How can I better express my results regarding $\displaystyle b$, and $\displaystyle c$?

Thanks!

2. Originally Posted by Mollier
Problem:
For which numbers $\displaystyle a,b,c,d$ will the function
$\displaystyle f(x)=\frac{ax+b}{cx+d}$
satisfy $\displaystyle f(f(x))=x$ for all $\displaystyle x$?
if $\displaystyle f[f(x)] = x$ , then $\displaystyle f(x)$ is its own inverse, and as such, its graph is symmetrical to the line $\displaystyle y = x$

$\displaystyle \frac{a \cdot f(x) + b}{c \cdot f(x) + d} = x$

$\displaystyle a \cdot f(x) + b = cx \cdot f(x) + dx$

$\displaystyle a \cdot f(x) - cx \cdot f(x) = dx - b$

$\displaystyle f(x)[a - cx] = dx - b$

$\displaystyle f(x) = \frac{dx-b}{a-cx} = \frac{ax+b}{cx+d}$

$\displaystyle \frac{-dx+b}{cx-a} = \frac{ax+b}{cx+d}$

the above equation only requires that $\displaystyle a = -d$.

3. Originally Posted by Mollier
Problem:
For which numbers $\displaystyle a,b,c,d$ will the function
$\displaystyle f(x)=\frac{ax+b}{cx+d}$
satisfy $\displaystyle f(f(x))=x$ for all $\displaystyle x$?

------------------------------------------------------------------------
Attempt:
$\displaystyle f(f(x))=\frac{a\left[\frac{ax+b}{cx+d}\right]+b}{c\left[\frac{ax+b}{cx+d}\right]+d}\\ = \frac{\left[\frac{a^2x+ab}{cx+d}\right]+b}{\left[\frac{acx+bc}{cx+d}\right]+d}\\ =\frac{(a^2+bc)x+(ab+bd)}{(ac+cd)x+(bc+d^2)}$ Correct up to here.

By using polynomial division I get the following result:

$\displaystyle \left(\frac{a^2+bc}{ac+cd}\right)+\frac{(ab+bd)-(bc+d^2)\left(\frac{a^2+bc}{ac+cd}\right)}{(ac+cd) x+(bc+d^2)}$ The trouble with this is that you are about to find that a = –d, so the denominator of the fraction in parentheses is zero.

By doing some hopefully correct algebra I get:

$\displaystyle (a^2+bc)x+(ab+bd)=x$
1. $\displaystyle (ab+bd)=0$
So, $\displaystyle a=-d$

2. $\displaystyle (a^2+bc)=1$
So, $\displaystyle a=\sqrt{1-bc}$

If $\displaystyle b$ and $\displaystyle c$ have opposite signs, their product is negative, and they can be any number.

If on the other hand, their signs are the same, their product is positive and:
$\displaystyle -1 \leq b \leq 1$
$\displaystyle -1 \leq c \leq 1$
-----------------------------------------------------------------------

Am I making this more complicated than it is?
How can I better express my results regarding $\displaystyle b$, and $\displaystyle c$?
The condition f(f(x)) tells you that $\displaystyle \frac{(a^2+bc)x+(ab+bd)}{(ac+cd)x+(bc+d^2)} = x$. Cross-multiply to get $\displaystyle (a^2+bc)x+(ab+bd) = x\bigl((ac+cd)x+(bc+d^2)\bigr)$. If that quadratic equation holds for all x then you can equate each coefficient to zero. You should find that there are two sets of solutions:

1. $\displaystyle a+d=0$ (b and c can be anything, so long as you don't have all four of a, b, c, d equal to 0);

2. $\displaystyle a = d \ne0,\;b=c=0$.

4. Originally Posted by skeeter
if [tex]
the above equation only requires that $\displaystyle a = -d$.
The way you come to that conclusion is very understandable, thank you.

Originally Posted by Opalg

$\displaystyle \left(\frac{a^2+bc}{ac+cd}\right)+\frac{(ab+bd)-(bc+d^2)\left(\frac{a^2+bc}{ac+cd}\right)}{(ac+cd) x+(bc+d^2)}$

The trouble with this is that you are about to find that a = –d, so the denominator of the fraction in parentheses is zero.
Ah, I did not see that! Thank you!

Merry Christmas