Given g(x) = x+5 and g(f(x)) = (6(x−3))/(x-4), find f(x).

We are given two functions:

$$g(x) = x+5$$

And the composite function:

$$g(f(x)) = \frac{6(x-3)}{x-4}$$

The notation $g(f(x))$ means that we substitute $f(x)$ into the function $g(x)$ wherever $x$ appears.

Using the definition of $g(x)$, replace $x$ with $f(x)$:

$$g(f(x)) = f(x) + 5$$

Now we have two expressions for $g(f(x))$, so we can set them equal to each other:

$$f(x) + 5 = \frac{6(x-3)}{x-4}$$

Subtract 5 from both sides of the equation:

$$f(x) = \frac{6(x-3)}{x-4} - 5$$

To combine the terms, find a common denominator, which is $(x-4)$:

$$f(x) = \frac{6(x-3)}{x-4} - \frac{5(x-4)}{x-4}$$

Combine the numerators:

$$f(x) = \frac{6(x-3) - 5(x-4)}{x-4}$$

Expand the terms in the numerator:

$$f(x) = \frac{6x - 18 - (5x - 20)}{x-4}$$

Distribute the negative sign:

$$f(x) = \frac{6x - 18 - 5x + 20}{x-4}$$

Combine like terms in the numerator:

$$f(x) = \frac{(6x - 5x) + (-18 + 20)}{x-4}$$ $$f(x) = \frac{x + 2}{x-4}$$

The value of $f(x)$ is: