The functions are given as,

Substitute the values of f(x) and g(x) in h(x), and simplify,

Thus, option C is the correct choice as it represents the correct expression of the function h(x).
Consider the given function g(x) as,
![g(x)=-\sqrt[]{x+3}+2](https://img.qammunity.org/2023/formulas/mathematics/college/6o74el0eeees7ty5dqj35rk972e6wbwbs9.png)
Since the function f(x) is not given here, we will use the same function from the above part of the problem,

According to the given condition,

Substitute the values,
![\begin{gathered} 3x+2=-\sqrt[]{x+3}+2 \\ 3x=-\sqrt[]{x+3} \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/hpamz7gk9820f3rg6ckwdqiefctp87phs7.png)
Consider this as the parent equation.
Square both sides,

Apply the quadratic formula,
![\begin{gathered} x=\frac{-(-1)\pm\sqrt[]{(-1)^2-4(9)(-3)}}{2(9)} \\ x=\frac{1\pm\sqrt[]{1+108}}{18} \\ x=\frac{1+\sqrt[]{109}}{18},\frac{1-\sqrt[]{109}}{18} \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/5rssndizbdg4rpyrov4v6tzo4jm3j59g5o.png)
Consider that the square root term cannot contain any negative number inside. So the following solution is not feasible,
![x=\frac{1-\sqrt[]{109}}{18}](https://img.qammunity.org/2023/formulas/mathematics/college/2dz2vyr3x9819aeooqnm12z287z92s1icn.png)
So, the only feasible solution of the parent equation is,
![x=\frac{1+\sqrt[]{109}}{18}](https://img.qammunity.org/2023/formulas/mathematics/college/cwi2o7i7ud2g4vc11rz5pkhfeahtd49e9c.png)
Thus, there is only one value of 'x' that satisfies the condition.
Therefore, option B is the correct choice.