Question

Given `f(x) = (3-2x) / (2x+1)` and `f(g(x)) = 7 - 3x` how do you find g(x)?

Answer 1

`g(x) = (4 - 3x) / (-16 + 6x)`

Explanation:

`f(g(x))` can be computed by plugging `g(x)` for every occurence of `x` in `f(x)`.

Even though we don't know `g(x)` yet, we can still do this:

`f(x) = (3 - 2x) / (2x + 1) " "=> " " f(g(x)) = (3 - 2 g(x)) / (2 g(x) + 1)`

We also know that `f(g(x)) = 7 - 3x`, so we have:

`(3 - 2 g(x)) / (2 g(x) + 1) = 7 - 3x`

Let me write `g` instead of `g(x)` for better readability:

`(3 - 2g)/(2g + 1) = 7 - 3x`

Now, you need to solve this equation for `g`:

... multiply both sides with `(2g+1)`...

`<=> 3 - 2g = (7 - 3x) * (2g + 1)`

`<=> 3 - 2g = (7 - 3x) * 2g + (7 - 3x)`

Bring all products that include `g` to the left side and everything else to the right side.
So, subtract `(7 - 3x) * 2g` on both sides, and subtract `3` on both sides:

`<=> - 2g - (7 - 3x) * 2g = (7 - 3x) - 3`

... factorize `g` on the left side...

`<=> (-2 - 14 + 6x) * g = 4 - 3x`

`<=> (-16 + 6x) * g = 4 - 3x`

... divide both sides by `(-16 + 6x)`...

`<=> g = (4 - 3x)/(-16 + 6x) = (4 - 3x)/(2(-8 + 3x))`

Thus, we have

`g(x) = (4 - 3x) / (-16 + 6x)`

It might be a good idea to test if the calculation was correct. To do so, compute `f(g(x))`:

`f(g(x)) = f((4 - 3x) / (-16 + 6x))`

`= (3 - 2 * (4 - 3x) / (2(-8+ 3x)))/(2 * (4 - 3x) / (2(-8 + 3x)) + 1)`

`= (3 - (4 - 3x) / (-8 + 3x))/( (4 - 3x) /(-8+ 3x) + 1)`

`= ((3(-8 + 3x) - (4 - 3x))/(-8 + 3x)) / ((4 - 3x + (-8 + 3x))/(-8 + 3x))`

`= (3(-8 + 3x) - (4 - 3x)) / (4 - 3x + (-8 + 3x)) = (-28 +12x) / (-4)`

`= 7 - 3x`