Questions: Find f+g, f-g, fg, and f/g, Determine the domain for each function. f(x)=sqrt(x+9) ; g(x)=sqrt(x+2) (f+g)(x)= What is the domain of f+g ? A. The domain of f+g is - (Type your answer in interval notation.) B. The domain of f+g is 3. (Use a comma to separate answers as needed.) C. The domain of f+g is ∅. (f-g)(x)= What is the domain of f-g ? A. The domain of f-g is 3. (Use a comma to separate answers as needed.) B. The domain of f-g is . (Type your answer in interval notation.) C. The domain of f-g is ∅. (fg)(x)= What is the domain of fg ? A. The domain of fg is - (Use a comma to separate answers as needed.) B. The domain of fg is . (Type your answer in interval notation.) C. The domain of fg is ∅. (f/g)(x)= What is the domain of f/g? A. The domain of f/g is - (Use a comma to separate answers as needed.) B. The domain of f/g is . (Type your answer in interval notation.) C. The domain of f/g is ∅.

Find f+g, f-g, fg, and f/g, Determine the domain for each function. f(x)=sqrt(x+9) ; g(x)=sqrt(x+2) (f+g)(x)= What is the domain of f+g ? A. The domain of f+g is - (Type your answer in interval notation.) B. The domain of f+g is 3. (Use a comma to separate answers as needed.) C. The domain of f+g is ∅. (f-g)(x)= What is the domain of f-g ? A. The domain of f-g is 3. (Use a comma to separate answers as needed.) B. The domain of f-g is . (Type your answer in interval notation.) C. The domain of f-g is ∅. (fg)(x)= What is the domain of fg ? A. The domain of fg is - (Use a comma to separate answers as needed.) B. The domain of fg is . (Type your answer in interval notation.) C. The domain of fg is ∅. (f/g)(x)= What is the domain of f/g? A. The domain of f/g is - (Use a comma to separate answers as needed.) B. The domain of f/g is . (Type your answer in interval notation.) C. The domain of f/g is ∅.
Transcript text: Find $\mathrm{f}+\mathrm{g}, \mathrm{f}-\mathrm{g}, \mathrm{fg}$, and $\frac{\mathrm{f}}{\mathrm{g}}$, Determine the domain for each function. \[ \begin{array}{l} f(x)=\sqrt{x+9} ; g(x)=\sqrt{x+2} \\ (f+g)(x)=\square \end{array} \] What is the domain of $\mathrm{f}+\mathrm{g}$ ? A. The domain of $f+g$ is $\square$ - (Type your answer in interval notation.) B. The domain of $f+g$ is $\square$ 3. (Use a comma to separate answers as needed.) C. The domain of $f+g$ is $\varnothing$. \[ (f-g)(x)= \] What is the domain of $\mathrm{f}-\mathrm{g}$ ? A. The domain of $\mathrm{f}-\mathrm{g}$ is \{ $\square$ 3. (Use a comma to separate answers as needed.) B. The domain of $f-g$ is $\square$ . (Type your answer in interval notation.) C. The domain of $\mathrm{f}-\mathrm{g}$ is $\varnothing$. $(\mathrm{fg})(\mathrm{x})=$ $\square$ What is the domain of fg ? A. The domain of $f g$ is $\square$ - (Use a comma to separate answers as needed.) B. The domain of fg is $\square$ . (Type your answer in interval notation.) C. The domain of fg is $\varnothing$. \[ \left(\frac{\mathrm{f}}{\mathrm{g}}\right)(\mathrm{x})=\square \] What is the domain of $\frac{\mathrm{f}}{\mathrm{g}}$? A. The domain of $\frac{\mathrm{f}}{\mathrm{g}}$ is $\square$ - (Use a comma to separate answers as needed.) B. The domain of $\frac{\mathrm{f}}{\mathrm{g}}$ is $\square$ . (Type your answer in interval notation.) C. The domain of $\frac{\mathrm{f}}{\mathrm{g}}$ is $\varnothing$.
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Solution

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Solution Steps

To solve the given problem, we need to perform the following steps:

  1. Addition of Functions (f+g): Add the functions f(x) f(x) and g(x) g(x) .
  2. Subtraction of Functions (f-g): Subtract the function g(x) g(x) from f(x) f(x) .
  3. Multiplication of Functions (fg): Multiply the functions f(x) f(x) and g(x) g(x) .
  4. Division of Functions (f/g): Divide the function f(x) f(x) by g(x) g(x) .
  5. Determine the Domain: For each operation, determine the domain where the resulting function is defined.
Step 1: Define the Functions

Given: f(x)=x+9 f(x) = \sqrt{x+9} g(x)=x+2 g(x) = \sqrt{x+2}

Step 2: Calculate (f+g)(x) (f+g)(x)

(f+g)(x)=f(x)+g(x)=x+9+x+2 (f+g)(x) = f(x) + g(x) = \sqrt{x+9} + \sqrt{x+2}

Step 3: Determine the Domain of f+g f+g

For f(x) f(x) and g(x) g(x) to be defined, the expressions inside the square roots must be non-negative: x+9≥0  ⟹  x≥−9 x + 9 \geq 0 \implies x \geq -9 x+2≥0  ⟹  x≥−2 x + 2 \geq 0 \implies x \geq -2 The domain of f+g f+g is the intersection of these intervals: x≥−2 x \geq -2 In interval notation, this is: [−2,∞) \boxed{[-2, \infty)}

Step 4: Calculate (f−g)(x) (f-g)(x)

(f−g)(x)=f(x)−g(x)=x+9−x+2 (f-g)(x) = f(x) - g(x) = \sqrt{x+9} - \sqrt{x+2}

Step 5: Determine the Domain of f−g f-g

The domain of f−g f-g is the same as the domain of f+g f+g since the same conditions apply: [−2,∞) \boxed{[-2, \infty)}

Step 6: Calculate (fg)(x) (fg)(x)

(fg)(x)=f(x)⋅g(x)=x+9⋅x+2=(x+9)(x+2) (fg)(x) = f(x) \cdot g(x) = \sqrt{x+9} \cdot \sqrt{x+2} = \sqrt{(x+9)(x+2)}

Step 7: Determine the Domain of fg fg

The domain of fg fg is the same as the domain of f+g f+g and f−g f-g : [−2,∞) \boxed{[-2, \infty)}

Final Answer

  • (f+g)(x)=x+9+x+2 (f+g)(x) = \sqrt{x+9} + \sqrt{x+2}
  • The domain of f+g f+g is [−2,∞) \boxed{[-2, \infty)}
  • (f−g)(x)=x+9−x+2 (f-g)(x) = \sqrt{x+9} - \sqrt{x+2}
  • The domain of f−g f-g is [−2,∞) \boxed{[-2, \infty)}
  • (fg)(x)=(x+9)(x+2) (fg)(x) = \sqrt{(x+9)(x+2)}
  • The domain of fg fg is [−2,∞) \boxed{[-2, \infty)}
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