Answer:
ℝ - {(-2/3),(3/2)}
Step-by-step explanation:
We want the domain of f(g(x)). So, firstly, we have to find the domain for g(x) and, then, for f(g(x)).
- Domain of g(x): Since the expression is a fracion, we must exclude the values of x that make null the denominator. Hence,
![g(x)=\dfrac{x+5}{2x-3}\Longrightarrow 2x-3\neq 0\iff \boxed{x\neq\dfrac{3}{2}}](https://tex.z-dn.net/?f=g%28x%29%3D%5Cdfrac%7Bx%2B5%7D%7B2x-3%7D%5CLongrightarrow%202x-3%5Cneq%200%5Ciff%20%5Cboxed%7Bx%5Cneq%5Cdfrac%7B3%7D%7B2%7D%7D)
- Domain of f(g(x)): We'll find its expression:
![f(x) = \dfrac{3x-2}{x+1}\\\\f(g(x)) = \dfrac{3g(x)-2}{g(x)+1}\\\\f(g(x)) = \dfrac{3\cdot\dfrac{x+5}{2x-3}-2}{\dfrac{x+5}{2x-3}+1}=\dfrac{~~~\dfrac{3(x+5)-2(2x-3)}{2x-3}~~~}{\dfrac{(x+5)+(2x-3)}{2x-3}}\\\\f(g(x)) =\dfrac{3(x+5)-2(2x-3)}{(x+5)+(2x-3)}=\dfrac{3x+15-4x+6}{x+5+2x-3}\\\\\boxed{f(g(x)) =\dfrac{21-x}{3x+2}}](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%5Cdfrac%7B3x-2%7D%7Bx%2B1%7D%5C%5C%5C%5Cf%28g%28x%29%29%20%3D%20%5Cdfrac%7B3g%28x%29-2%7D%7Bg%28x%29%2B1%7D%5C%5C%5C%5Cf%28g%28x%29%29%20%3D%20%5Cdfrac%7B3%5Ccdot%5Cdfrac%7Bx%2B5%7D%7B2x-3%7D-2%7D%7B%5Cdfrac%7Bx%2B5%7D%7B2x-3%7D%2B1%7D%3D%5Cdfrac%7B~~~%5Cdfrac%7B3%28x%2B5%29-2%282x-3%29%7D%7B2x-3%7D~~~%7D%7B%5Cdfrac%7B%28x%2B5%29%2B%282x-3%29%7D%7B2x-3%7D%7D%5C%5C%5C%5Cf%28g%28x%29%29%20%3D%5Cdfrac%7B3%28x%2B5%29-2%282x-3%29%7D%7B%28x%2B5%29%2B%282x-3%29%7D%3D%5Cdfrac%7B3x%2B15-4x%2B6%7D%7Bx%2B5%2B2x-3%7D%5C%5C%5C%5C%5Cboxed%7Bf%28g%28x%29%29%20%3D%5Cdfrac%7B21-x%7D%7B3x%2B2%7D%7D)
Now, once again, we have to exclude the values of x that make the denominator equals to zero. Thus,
![f(g(x)) =\dfrac{21-x}{3x+2}\Longrightarrow 3x+2\neq0\iff \boxed{x\neq-\dfrac{2}{3}}](https://tex.z-dn.net/?f=f%28g%28x%29%29%20%3D%5Cdfrac%7B21-x%7D%7B3x%2B2%7D%5CLongrightarrow%203x%2B2%5Cneq0%5Ciff%20%5Cboxed%7Bx%5Cneq-%5Cdfrac%7B2%7D%7B3%7D%7D)
Lastly, we may write the domanin of f(g(x)):
![D(f(g(x)) = \left]-\infty,-\dfrac{2}{3}\right[\cup\left]-\dfrac{2}{3},\dfrac{3}{2}\right[\cup\left]\dfrac{3}{2},\infty\right[](https://tex.z-dn.net/?f=D%28f%28g%28x%29%29%20%3D%20%5Cleft%5D-%5Cinfty%2C-%5Cdfrac%7B2%7D%7B3%7D%5Cright%5B%5Ccup%5Cleft%5D-%5Cdfrac%7B2%7D%7B3%7D%2C%5Cdfrac%7B3%7D%7B2%7D%5Cright%5B%5Ccup%5Cleft%5D%5Cdfrac%7B3%7D%7B2%7D%2C%5Cinfty%5Cright%5B)
or, just writing in a shorter way:
![\boxed{D(f(g(x)) = \mathbb{R}-\left\{-\dfrac{2}{3},\dfrac{3}{2}\right\}}](https://tex.z-dn.net/?f=%5Cboxed%7BD%28f%28g%28x%29%29%20%3D%20%5Cmathbb%7BR%7D-%5Cleft%5C%7B-%5Cdfrac%7B2%7D%7B3%7D%2C%5Cdfrac%7B3%7D%7B2%7D%5Cright%5C%7D%7D)