A volunteer tutor wants to know HOW to solve this problem. It is not
necessary to give the answer if you have explained it so that a
student can get it...
Function f is defined as y=f(x)=(2x+1)/(x-3) where x does not equal 3.
Find the value of k so that (f^-1)(x)=(3x+1)/(x-k).

Hi gyrocopter,
I'd start by finding the (f^-1)(x) function first and ignoring the
'find the value of k so that...' part for the time being.

Let (f^-1)(x) = a

f(a) = x

(2a+1)/(a-3) = x

2a+1 = ax-3x

2a-ax = -3x-1

a(2-x) = -3x-1

a = (3x+1)/(x-2) = (f^-1)(x)

So, k = 2.


In case the student is unfamiliar with inverse functions, I'd
recommend this website.
"The inverse of a function has all the same points, except that the
x's and y's have been reversed."
http://www.purplemath.com/modules/invrsfcn.htm

If you need any clarification, just ask.

Search strategy
"f inverse x"

thanks for the comment too

Some other possible approaches:

I.  Notice that in the definition of f(x), you said f(x)=(2x+1)/(x-3)
**where x does not equal 3**.  This is because the denominator becomes
undefined.  Note that similarly the denominator of (f^-1)(x) becomes
undefined when x=k.  Therefore, k is the one value that f(x) **does
not assume** (i.e., is not in the range of f).  This is because if
f(x)=y, then (f^-1)(y)=x, so (f^-1) must be defined on all of the
values that f assumes.

The function f's asymptote, however, is y=2, and f never assumes the
exact value y=2.  Hence k=2.  [Note that if you try to solve f(x)=2,
you get an impossibility such as 1=-6.]

II.  Plug in a value of x into y=f(x).  For example, try x=1 and get
y=-3/2.  Then, when we plug in x=-3/2 into (f^-1)(x), we should get 1.
 So,
  1 = (3*(-3/2)+1)/(-3/2 - k) = (-7/2) / (-3/2 - k).
Solving for k gives k = -3/2 + 7/2 = 2, as before.

Making a smart choice for the original x can simplify the problem.  In
particular, plugging in x=-1/2 gives y=0, so plugging x=0 into
(f^-1)(x) should give -1/2.  This leads to the equation -1/2 = 1/(-k)
which immediately gives k=2.