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

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

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

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

Step 3: Determine the Domain of \( f+g \)

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

Step 4: Calculate \( (f-g)(x) \)

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

Step 5: Determine the Domain of \( f-g \)

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

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

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

Step 7: Determine the Domain of \( fg \)

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

Final Answer

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