QuestionGiven the following functions: - $f(x)=\frac{4 x-3}{x-2}$ - $g(x)=2 x-1$
Determine the rule of $g \circ f(x+h)$.

Studdy Solution
Simplify the expression for $g(f(x+h))$:
$$g(f(x+h)) = \frac{2(4x + 4h - 3)}{x + h - 2} - 1$$
$$g(f(x+h)) = \frac{8x + 8h - 6}{x + h - 2} - 1$$
To combine into a single fraction:
$$g(f(x+h)) = \frac{8x + 8h - 6 - (x + h - 2)}{x + h - 2}$$
$$g(f(x+h)) = \frac{8x + 8h - 6 - x - h + 2}{x + h - 2}$$
$$g(f(x+h)) = \frac{7x + 7h - 4}{x + h - 2}$$
The rule of $g \circ f(x+h)$ is $\boxed{\frac{7x + 7h - 4}{x + h - 2}}$.