college Algebra

Question

Determine whether the function is one-to-one. (If it is not, provide an example of an two different x-values giving the same y-value).

 

F(x)= 3x/x-2

Answer 1

f(x)=3x/(x-2) = (3(x-2))/(x-2)+6/(x-2)=3+6/(x-2)

This graph is the graph y=1/x, stretched vertically by a factor of 6, then pushed up 3 units.  The graph y=1/x is 1-1.  Borrowing Jon's notation

If 1/x=1/P then multiplying each side by xP we have P=x.

Answer 2

I assume you mean 3x / (x - 2)

 

You can figure out if there are any x values giving the same y value by starting with the assumption that there are and seeing what you get.

 

Pick some arbitrary x value -- call it P.  So y in this case would be equal to 3P / (P-2).

 

Let's suppose there is some other x value that has the same y value.  Let's see if we can figure out what it is.

 

F(x) would have to be equal to 3P / (P-2).  So 3x / (x - 2) = 3P / (P-2).  Let's see what happens if we try to solve for x in that case.  (Certainly x = P would work, but we're trying to see if there's another x that would also work.)

 

3x / (x - 2) = 3P / (P - 2)

3x = (x - 2) (3P) / (P - 2)

3x = 3Px / (P - 2) - 6P / (P - 2)

3x = 3x P / (P - 2) - 6P / (P - 2)

3x - 3x P / (P - 2) = - 6P / (P - 2)

3x (1 - P / (P - 2)) = - 6P / (P - 2)

3x ((P - 2 - P) / (P - 2) = - 6P / (P - 2)

3x (-2 / (P - 2)) = - 6P / (P - 2)

3x (-2) = -6P

-6x = -6P

x = P

 

That may seem like a long way around to get to x = P, but what it shows is that the only value of x that gives a Y value of 3P / (P-2) is P itself.  There is no other x value that gives the same y value.  And since P was chosen completely arbitrarily, that point holds for all P.

 

So there is only one value of x that gives a specific Y value, and therefore the function is one-to-one.