Find $$ f+g, f-g, f g, \frac{f}{g} $$. Determine the domain for each function. $$ f(x)=\frac{9 x}{x-3}, g(x)=\frac{7}{x+7} $$

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Let's compute each of the requested operations between the functions $$ f(x) = \frac{9x}{x-3} $$ and $$ g(x) = \frac{7}{x+7} $$. We'll also determine the domain for each resulting function.

### 1. Sum: $$ f + g $$

$$
f + g = \frac{9x}{x - 3} + \frac{7}{x + 7}
$$

To add these fractions, find a common denominator:

$$
f + g = \frac{9x(x + 7) + 7(x - 3)}{(x - 3)(x + 7)} = \frac{9x^2 + 63x + 7x - 21}{(x - 3)(x + 7)} = \frac{9x^2 + 70x - 21}{(x - 3)(x + 7)}
$$

**Domain:** $$ x \neq 3 $$ and $$ x \neq -7 $$

---

### 2. Difference: $$ f - g $$

$$
f - g = \frac{9x}{x - 3} - \frac{7}{x + 7}
$$

Find a common denominator to subtract the fractions:

$$
f - g = \frac{9x(x + 7) - 7(x - 3)}{(x - 3)(x + 7)} = \frac{9x^2 + 63x - 7x + 21}{(x - 3)(x + 7)} = \frac{9x^2 + 56x + 21}{(x - 3)(x + 7)}
$$

**Domain:** $$ x \neq 3 $$ and $$ x \neq -7 $$

---

### 3. Product: $$ f \cdot g $$

$$
f \cdot g = \frac{9x}{x - 3} \times \frac{7}{x + 7} = \frac{63x}{(x - 3)(x + 7)}
$$

**Domain:** $$ x \neq 3 $$ and $$ x \neq -7 $$

---

### 4. Quotient: $$ \frac{f}{g} $$

$$
\frac{f}{g} = \frac{\frac{9x}{x - 3}}{\frac{7}{x + 7}} = \frac{9x}{x - 3} \times \frac{x + 7}{7} = \frac{9x(x + 7)}{7(x - 3)} = \frac{9x^2 + 63x}{7(x - 3)}
$$

**Domain:** $$ x \neq 3 $$ and $$ x \neq -7 $$

---

### Summary of Results

1. **Sum:**
   $$
   f + g = \frac{9x^2 + 70x - 21}{(x - 3)(x + 7)}, \quad \text{Domain: } x \neq 3, -7
   $$

2. **Difference:**
   $$
   f - g = \frac{9x^2 + 56x + 21}{(x - 3)(x + 7)}, \quad \text{Domain: } x \neq 3, -7
   $$

3. **Product:**
   $$
   f \cdot g = \frac{63x}{(x - 3)(x + 7)}, \quad \text{Domain: } x \neq 3, -7
   $$

4. **Quotient:**
   $$
   \frac{f}{g} = \frac{9x^2 + 63x}{7(x - 3)}, \quad \text{Domain: } x \neq 3, -7
   $$

All resulting functions are defined for all real numbers except $$ x = 3 $$ and $$ x = -7 $$.
Here are the results for each operation between $$ f(x) = \frac{9x}{x-3} $$ and $$ g(x) = \frac{7}{x+7} $$, along with their domains: 1. **Sum ($$ f + g $$):** $$ f + g = \frac{9x^2 + 70x - 21}{(x - 3)(x + 7)} $$ *Domain:* $$ x \neq 3 $$ and $$ x \neq -7 $$ 2. **Difference ($$ f - g $$):** $$ f - g = \frac{9x^2 + 56x + 21}{(x - 3)(x + 7)} $$ *Domain:* $$ x \neq 3 $$ and $$ x \neq -7 $$ 3. **Product ($$ f \cdot g $$):** $$ f \cdot g = \frac{63x}{(x - 3)(x + 7)} $$ *Domain:* $$ x \neq 3 $$ and $$ x \neq -7 $$ 4. **Quotient ($$ \frac{f}{g} $$):** $$ \frac{f}{g} = \frac{9x^2 + 63x}{7(x - 3)} $$ *Domain:* $$ x \neq 3 $$ and $$ x \neq -7 $$ All functions are undefined at $$ x = 3 $$ and $$ x = -7 $$.

Find \( f+g, f-g, f g, \frac{f}{g} \). Determine the domain for each function. \( f(x)=\frac{9 x}{x-3}, g(x)=\frac{7}{x+7} \)

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