Function `f(x) = {3x-1}/{x-2}`, x not equal to 2 and `f(x) = {2x-1}/{x-3}`, x not equal to 3, show that fog(x) = x

Solution:

To show that fog(x) = x, we need to find the composition of the two functions f(x) and g(x) and demonstrate that it equals x.

Let's first find the expression for g(x):

`g(x) = {2x-1}/{x-3}, x ≠ 3`

Now, let's find the expression for f(g(x)) by substituting g(x) into f(x):

`f(g(x)) = f({2x-1}/{x-3})`

Using the definition of f(x), we substitute `{2x-1}/{x-3}` into f(x):

`f(g(x)) = {[(3 * ({2x-1}/{x-3})) - 1]}/{(({2x-1}/{x-3}) - 2)}`

Now, we simplify the expression:

`f(g(x)) = {[{6x - 3}/{x - 3} - 1]}/{({2x - 1 - 2x + 6}/{x - 3})}`

`f(g(x)) = {[{6x - 3}/{x - 3} - 1]}/{{5}/{x - 3}}`

To further simplify, let's multiply the numerator by (x - 3) and simplify:

`f(g(x)) = {[{6x - 3 - (x - 3)}/{x - 3}]} / {{5}/{x - 3}}`

`f(g(x)) = {[{6x - 3 - x + 3}/{x - 3}]} / {{5}/{x - 3}}`

`f(g(x)) = {[{5x}/{x - 3}] }/ {{5}/{x - 3}}`

Now, we can simplify by canceling out the (x - 3) terms:

`f(g(x)) = {5x}/{5}`

`f(g(x)) = x`

Therefore, we have shown that fog(x) = x.