Given \( f \) and \( g \) as defined below, determine \( f+g, f-g, f g \), and \( \frac{f}{g} \). Using interval notation, report the domain of each result. \( f(x)=x^{2}-5 x \) and \( g(x)=x+6 \) (a) \( (f+g)(x)=\square \) Domain of \( (f+g)(x): \) (b) \( (f-g)(x)=\square \) Domain of \( (f-g)(x): \) (c) \( (f g)(x)=\square \) Domain of \( (f g)(x): \) (d) \( \left(\frac{f}{g}\right)(x)=\square \) Duestion Help: \( (x) \)

To find \( f+g \), \( f-g \), \( fg \), and \( \frac{f}{g} \), we will first define the functions \( f(x) \) and \( g(x) \): \[ f(x) = x^2 - 5x \] \[ g(x) = x + 6 \] Now, let's compute each operation step by step. ### (a) \( (f+g)(x) \) \[ (f+g)(x) = f(x) + g(x) = (x^2 - 5x) + (x + 6) \] \[ = x^2 - 5x + x + 6 = x^2 - 4x + 6 \] **Domain of \( (f+g)(x) \):** The domain of \( f(x) \) is all real numbers, and the domain of \( g(x) \) is also all real numbers. Therefore, the domain of \( (f+g)(x) \) is: \[ \text{Domain of } (f+g)(x): \quad (-\infty, \infty) \] ### (b) \( (f-g)(x) \) \[ (f-g)(x) = f(x) - g(x) = (x^2 - 5x) - (x + 6) \] \[ = x^2 - 5x - x - 6 = x^2 - 6x - 6 \] **Domain of \( (f-g)(x) \):** Similar to the previous case, the domain of \( (f-g)(x) \) is: \[ \text{Domain of } (f-g)(x): \quad (-\infty, \infty) \] ### (c) \( (fg)(x) \) \[ (fg)(x) = f(x) \cdot g(x) = (x^2 - 5x)(x + 6) \] Using the distributive property: \[ = x^2(x + 6) - 5x(x + 6) = x^3 + 6x^2 - 5x^2 - 30x \] \[ = x^3 + (6x^2 - 5x^2) - 30x = x^3 + x^2 - 30x \] **Domain of \( (fg)(x) \):** The domain of \( (fg)(x) \) is also all real numbers: \[ \text{Domain of } (fg)(x): \quad (-\infty, \infty) \] ### (d) \( \left(\frac{f}{g}\right)(x) \) \[ \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{x^2 - 5x}{x + 6} \] **Domain of \( \left(\frac{f}{g}\right)(x) \):** The function \( g(x) \) cannot be zero, so we need to find where \( g(x) = 0 \): \[ x + 6 = 0 \implies x = -6 \] Thus, the domain excludes \( x = -6 \): \[ \text{Domain of } \left(\frac{f}{g}\right)(x): \quad (-\infty, -6) \cup (-6, \infty) \] ### Summary of Results (a) \( (f+g)(x) = x^2 - 4x + 6 \) Domain: \( (-\infty, \infty) \) (b) \( (f-g)(x) = x^2 - 6x - 6 \) Domain: \( (-\infty, \infty) \) (c) \( (fg)(x) = x^3 + x^2 - 30x \) Domain: \( (-\infty, \infty) \) (d) \( \left(\frac{f}{g}\right)(x) = \frac{x^2 - 5x}{x + 6} \) Domain: \( (-\infty, -6) \cup (-6, \infty) \)