Let $$ f(x)=4 x+5 $$ and $$ g(x)=2 x-7 $$. Find $$ (f+g)(x) $$, $$ (f-g)(x) $$, $$ (f g)(x) $$, and $$ \left(\frac{1}{g}\right)(x) $$. Give the domain of each. $$ (f+g)(x)=\square $$ (Simplify your answer.) $$ (f-g)(x)=\square $$ (Simplify your answer.) $$ \left(\frac{f g}{}\right)(x)=\square $$ (Simplify your answer.) $$ \left(\frac{f}{g}\right)(x)=\square $$ (Simplify your answer.) The domain of $$ f+g $$ is $$ \square $$. (Type your answer in interval notation.) The domain of $$ f-g $$ is $$ \square $$. (Type your answer in interval notation.) The domain of $$ f g $$ is $$ \square $$. (Type your answer in interval notation.)

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UpStudy AI · Accepted Answer

We are given:
  f(x) = 4x + 5  
  g(x) = 2x – 7

1. (f + g)(x):
  (f + g)(x) = f(x) + g(x)  
  = (4x + 5) + (2x – 7)  
  = 4x + 2x + 5 – 7  
  = 6x – 2

Domain of f + g: Since both f and g are defined for all real numbers, the domain is (–∞, ∞).

2. (f – g)(x):
  (f – g)(x) = f(x) – g(x)  
  = (4x + 5) – (2x – 7)  
  = 4x + 5 – 2x + 7  
  = 2x + 12

Domain of f – g: Again, since both f and g are defined for all real numbers, the domain is (–∞, ∞).

3. (fg)(x):
  (fg)(x) = f(x) · g(x)  
  = (4x + 5)(2x – 7)  
  Multiply using distributive property:
  = 4x(2x) + 4x(–7) + 5(2x) + 5(–7)  
  = 8x² – 28x + 10x – 35  
  = 8x² – 18x – 35

Domain of fg: Since multiplying two functions defined for all real numbers still has domain (–∞, ∞), the domain is (–∞, ∞).

4. (f/g)(x):
  (f/g)(x) = f(x) / g(x)  
  = (4x + 5) / (2x – 7)

Domain of f/g: The expression is undefined where the denominator equals 0:
  Set 2x – 7 = 0 ⟹ x = 7/2  
  Thus, the domain is all real numbers except x = 7/2, which in interval notation is (–∞, 7/2) ∪ (7/2, ∞).

Summary of answers:

(f + g)(x) = 6x – 2  
  (f – g)(x) = 2x + 12  
  (fg)(x) = 8x² – 18x – 35  
  (f/g)(x) = (4x + 5)/(2x – 7)

Domain of f + g: (–∞, ∞)  
  Domain of f – g: (–∞, ∞)  
  Domain of fg: (–∞, ∞)  
  Domain of f/g: (–∞, 7/2) ∪ (7/2, ∞)
$$ (f+g)(x) = 6x - 2 $$ $$ (f-g)(x) = 2x + 12 $$ $$ (fg)(x) = 8x^2 - 18x - 35 $$ $$ \left(\frac{f}{g}\right)(x) = \frac{4x + 5}{2x - 7} $$ Domain of $$ f+g $$: $$ (-\infty, \infty) $$ Domain of $$ f-g $$: $$ (-\infty, \infty) $$ Domain of $$ fg $$: $$ (-\infty, \infty) $$ Domain of $$ \frac{f}{g} $$: $$ (-\infty, \frac{7}{2}) \cup (\frac{7}{2}, \infty) $$

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