Question

Question #a678e

Answer 1

Start with an understanding of an inverse function...

Explanation:

...understand that an inverse function (call it g(x)) meets the condition:

`f(g(x)) = x`

...so we can just plug g(x) into the formula we are given for f(x), and then set that equal to x, and then algebraically solve for it. It will look like:

`(2(g(x)) + 1)/(4(g(x)) - 1) = x`

...multiply both sides by`(4(g(x))-1)`, giving:

`2(g(x)) + 1 = x(4(g(x))-1) = 4x(g(x))-x`

...we want to get the terms involving g(x) all on the same side,
so we subtract 2(g(x)) from both sides:

`1 = 4x(g(x))-x - 2(g(x))`

we want terms NOT involving g(x) all on the same side, so we add `x` to both sides:

`1 + x = 4x(g(x))- 2(g(x))`

...now we can factor out g(x) on the right side:

`1 + x = g(x)(4x -2)`

...and now we can divide both sides by (4x-2), and we've got it!

`(1+x)/(4x-2) = g(x)`

...check your work when you can. Yeah, I know, it's double the work, but it will save you on occasion, and will show your instructor that you care. He'll write you a nice recommendation or something.

We check this result by plugging in our derived function g(x) into the original function definition, and show that f(g(x)) = x.

`(2((1+x)/(4x-2))+1)/(4((1+x)/(4x-2)) - 1)`

`=((2 + 2x)/(4x-2) + (4x-2)/(4x-2))/((4 +4x)/(4x-2) -(4x-2)/(4x-2))`

`=((6x)/(4x-2))/(6/(4x-2)) = ((6x)/(4x-2)) * ((4x-2)/6)`

`=(6x)/6 = x`

...they GOTTA be happy with that!

GOOD LUCK!