To find \((f+g)(x)\), we first need to separately write down the functions \(f(x)\) and \(g(x)\).
The given functions are:
- \(f(x) = -2x^2 + 4x - 7\)
- \(g(x) = -\sqrt[3]{x - 1} + 2\)
Now, we add these two functions together:
\[ (f+g)(x) = f(x) + g(x) = (-2x^2 + 4x - 7) + \left(-\sqrt[3]{x - 1} + 2\right) \]
Combining the constants: \[ (f+g)(x) = -2x^2 + 4x - 7 + 2 - \sqrt[3]{x - 1} \]
Simplifying further: \[ (f+g)(x) = -2x^2 + 4x - 5 - \sqrt[3]{x - 1} \]
Next, we consider the domain of \((f+g)(x)\):
- The function \(-2x^2 + 4x - 5\) is a polynomial and defined for all real numbers.
- The term \(-\sqrt[3]{x - 1}\) is also defined for all real numbers, as cube root functions are defined for any real number.
Thus, the overall function \((f+g)(x)\) is defined for all real numbers.
So, the final expressions are: \[ (f+g)(x) = -2x^2 + 4x - 5 - \sqrt[3]{x - 1} \] and the domain of \((f+g)(x)\) is: \[ (-\infty, \infty) \]
Based on the options provided, the correct choice seems to be: \[ (f+g)(x)=-2x^2+4x-\sqrt[3]{x-1}-5 \quad \text{and the domain of } (f+g)(x) \text{ is } (-\infty, \infty). \]