Question

refer to the functions r and p. find the function (r - p)(x) and write the domain in interval notation.\\( r(x) = -4x \\)\\( p(x) = x^2 + 7x \\)\\( q(x) = \sqrt{5 - x} \\)\
part 1 of 2\\( (r - p)(x) = -x^2 - 11x \\)\
part: 1 / 2\
part 2 of 2\
the domain of (r - p)(x) is \\( \square \\).

Explanation:

Step1: Analyze the functions \( r(x) \) and \( p(x) \)

The function \( r(x) = -4x \) is a linear function, and its domain is all real numbers (\( (-\infty, \infty) \)) because there are no restrictions (like square roots, denominators, etc.) that would limit the input values. The function \( p(x) = x^2 + 7x \) is a quadratic function, and its domain is also all real numbers since polynomials are defined for all real numbers.

Step2: Determine the domain of \( (r - p)(x) \)

The function \( (r - p)(x) \) is defined as \( r(x) - p(x) \). When we subtract two functions, the domain of the resulting function is the intersection of the domains of the individual functions. Since the domain of \( r(x) \) is \( (-\infty, \infty) \) and the domain of \( p(x) \) is \( (-\infty, \infty) \), the intersection of these two domains (which is the domain of \( (r - p)(x) \)) is also \( (-\infty, \infty) \).

Answer:

\( (-\infty, \infty) \)