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lian · Feb 28, 2007, 07:07 PM

f(x)=x+2/x+6
what's the inverse

Ashes92 · Feb 28, 2007, 07:11 PM

Originally Posted by lian

f(x)=x+2/x+6
whats the inverse

I believe its x+2/x+6=f(x)

terryg752 · Oct 8, 2007, 06:45 AM

Originally Posted by lian

f(x)=x+2/x+6
whats the inverse

I don't think it has an inverse!

Inverse is possible only if the function is 1-1

But when y=9, we have 2 values of x: 1 and 2

asterisk_man · Oct 8, 2007, 07:44 AM

Disagree with both answers.
Ashes92 doesn't know what an inverse is and terryg752... I just don't know where you're coming from. y=9 only when x=-6.5. If you look at the graph there's a vertical asymptote at y=-6 and horizontal at x=1. There is no y value which occurs at multiple x values.

The correct way to solve is as follows:
$ f(x)=\frac{x+2}{x+6} \\ \text{replace f(x) with y for convenience} \\ y=\frac{x+2}{x+6} \\ \text{multiply both sides by x+6} \\ xy+6y=x+2 \\ \text{move all the x to the left and the non-x to the right} \\ xy-x=2-6y \\ \text{factor out the x on the left} \\ x\left({y-1}\right) =2-6y \\ \text{divide by y-1} \\ x=\frac{2-6y}{y-1} \\ \text{replace x with f-1 and y with x so we have the correct symbols for the inverse function}\\ f^{-1}(x)=\frac{2-6x}{x-1} $
now a quick test to see if we're right.
remember that (f-1(f(x))=x)
f(10)=12/16
f-1(12/16)=10
so we've got the right answer.

terryg752 · Oct 8, 2007, 04:12 PM

Originally Posted by lian

f(x)=x+2/x+6
whats the inverse

I thought you did not mean: (x+2)/(x+6) But X +2/X +6.

If you meant the former, you should have put brackets.

In the latter case there is no inverse as I stated earlier.

If you meant y = (x+2)/(x+6) THEN:

y(x+6) = x + 2

x(y-1) = 2 - 6 y

x = (2-6y)/(y-1)

So: Inverse function is (2-6x)/((x-1)