Find \( f+g, f-g, f g \), and \( f / g \) and their domains. \[ f(x)=x^{2}+x, \quad g(x)=x^{2} \] Find \( (f+g)(x) \). \( \square \) Find the domain of \( (f+g)(x) \). (Enter your answer using interval notation.) \( \square \) Find \( (f-g)(x) \). \( \square \) Find the domain of \( (f-g)(x) \). (Enter your answer using interval notation.) \( \square \) Find \( (f g)(x) \). \( \square \) Find the domain of \( (f g)(x) \). (Enter your answer using interval notation.) \( \square \) Find \( \left(\frac{f}{g}\right)(x) \). \( \square \) Find the domain of \( \left(\frac{f}{g}\right)(x) \). (Enter your answer using interval notation.) \( \square \)

We are given:   f(x) = x² + x   g(x) = x² Below are the steps to find the required functions and their domains. ────────────────────────────── 1. (f + g)(x):   (f + g)(x) = f(x) + g(x)          = (x² + x) + x²          = 2x² + x Domain of (f + g)(x): Since f and g are both polynomials (which are defined for all real numbers), the domain is   (–∞, ∞). ────────────────────────────── 2. (f – g)(x):   (f – g)(x) = f(x) – g(x)          = (x² + x) – x²          = x Domain of (f – g)(x): Again, being a polynomial, its domain is   (–∞, ∞). ────────────────────────────── 3. (f · g)(x):   (fg)(x) = f(x) · g(x)          = (x² + x) · x²          = x²(x² + x)          = x⁴ + x³ Domain of (f · g)(x): As a polynomial, the domain is   (–∞, ∞). ────────────────────────────── 4. (f / g)(x):   (f / g)(x) = f(x) / g(x)          = (x² + x) / x² Before simplifying, note that division by zero is not allowed. Since g(x) = x², we must have x² ≠ 0, i.e. x ≠ 0. Now, simplify:   f(x) / g(x) = (x² + x) / x² = [x(x + 1)] / x² = (x + 1) / x      (for x ≠ 0) Domain of (f / g)(x): x can be any real number except 0, so the domain is:   (–∞, 0) ∪ (0, ∞). ────────────────────────────── Summary of Answers: 1. (f + g)(x) = 2x² + x   Domain: (–∞, ∞) 2. (f – g)(x) = x   Domain: (–∞, ∞) 3. (f · g)(x) = x⁴ + x³   Domain: (–∞, ∞) 4. (f / g)(x) = (x + 1)/x  (for x ≠ 0)   Domain: (–∞, 0) ∪ (0, ∞)