The functions \( f \) and \( g \) are defined as \( f(x)=8 x+5 \) and \( g(x)=7-9 x \). a) Find the domain of \( f, g, f+g, f-g, f g, f f, \frac{f}{g} \), and \( \frac{g}{f} \). b) Find \( (f+g)(x),(f-g)(x),(f g)(x),(f f)(x),\left(\frac{f}{g}\right)(x) \), and \( \left(\frac{g}{f}\right)(x) \). a) The domain of \( f \) is \( \square \). (Type your answer in interval notation.)

We are given   f(x) = 8x + 5  and  g(x) = 7 – 9x. Below, we first determine the domains of the various functions, and then we compute the combined functions. ───────────────────────────── Part (a): Domains 1. Domain of f:   Since f(x) = 8x + 5 is a linear function, it is defined for every real number.   Domain: (–∞, ∞) 2. Domain of g:   g(x) = 7 – 9x is also linear, so it is defined for every real number.   Domain: (–∞, ∞) 3. Domain of (f + g) and (f – g):   Since f+g and f–g are sums/differences of two linear (hence continuous) functions, they are defined for all real numbers.   Domain for both: (–∞, ∞) 4. Domain of (f · g):   The product of two linear functions is a polynomial, which is defined for all real numbers.   Domain: (–∞, ∞) 5. Domain of (f f) (interpreted as f composed with f, i.e. f(f(x))):   Since f maps all real numbers to all real numbers, the composition f∘f is defined for every real x.   Domain: (–∞, ∞) 6. Domain of (f/g):   This is a rational function. The only restrictions occur when the denominator is zero.   Solve g(x) = 0:     7 – 9x = 0 ⟹ x = 7/9.   Thus, (f/g)(x) is defined for all real numbers except x = 7/9.   Domain: (–∞, 7/9) ∪ (7/9, ∞) 7. Domain of (g/f):   Similarly, for (g/f)(x), we need f(x) ≠ 0.   Solve f(x) = 0:     8x + 5 = 0 ⟹ x = –5/8.   So, (g/f)(x) is defined for all x except x = –5/8.   Domain: (–∞, –5/8) ∪ (–5/8, ∞) ───────────────────────────── Part (b): Function Combinations 1. (f + g)(x):   = f(x) + g(x)   = (8x + 5) + (7 – 9x)   = 8x – 9x + 5 + 7   = –x + 12 2. (f – g)(x):   = f(x) – g(x)   = (8x + 5) – (7 – 9x)   = 8x + 5 – 7 + 9x   = 17x – 2 3. (f · g)(x):   = f(x)·g(x)   = (8x + 5)(7 – 9x)   Expand:     = 8x·7 + 8x·(–9x) + 5·7 + 5·(–9x)     = 56x – 72x² + 35 – 45x   Combine like terms:     = –72x² + 11x + 35 4. (f f)(x) (interpreted as f composed with f, f(f(x))):   = f(f(x))   = f(8x + 5)   = 8(8x + 5) + 5   = 64x + 40 + 5   = 64x + 45 5. (f/g)(x):   = [f(x)] / [g(x)]   = (8x + 5) / (7 – 9x)   (Remember: x ≠ 7/9.) 6. (g/f)(x):   = [g(x)] / [f(x)]   = (7 – 9x) / (8x + 5)   (Remember: x ≠ –5/8.) ───────────────────────────── Final Answers for Part (a): a) The domain of f is (–∞, ∞).