$$ f(x)=x-5, g(x)=4 x^{2} $$ Find $$ f+g, f-g, f g $$, and $$ \frac{f}{g} $$. Determine the domain for each function.

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### Step 1: Define the Functions
We have
$$
f(x) = x - 5 \quad \text{and} \quad g(x) = 4x^2.
$$

### Step 2: Find $$ f+g $$
Compute the sum:
$$
f(x) + g(x) = (x - 5) + 4x^2 = 4x^2 + x - 5.
$$

**Domain:**  
Since both $$ f(x) $$ and $$ g(x) $$ are defined for all real numbers, the domain is 
$$
\{x \in \mathbb{R}\}.
$$

### Step 3: Find $$ f-g $$
Compute the difference:
$$
f(x) - g(x) = (x - 5) - 4x^2 = -4x^2 + x - 5.
$$

**Domain:**  
Again, both functions are defined for all real numbers. Hence, the domain is 
$$
\{x \in \mathbb{R}\}.
$$

### Step 4: Find $$ f \cdot g $$
Compute the product:
$$
f(x) \cdot g(x) = (x - 5) \cdot 4x^2 = 4x^2 (x - 5) = 4x^3 - 20x^2.
$$

**Domain:**  
Both functions are defined for all real numbers. The domain is 
$$
\{x \in \mathbb{R}\}.
$$

### Step 5: Find $$ \frac{f}{g} $$
Compute the quotient:
$$
\frac{f(x)}{g(x)} = \frac{x - 5}{4x^2}.
$$

**Domain:**  
The quotient is defined when the denominator is not zero. Set
$$
4x^2 \neq 0 \quad \Longrightarrow \quad x \neq 0.
$$
Thus, the domain is 
$$
\{x \in \mathbb{R} : x \neq 0\}.
$$
Here are the results: 1. **$$ f + g $$:** $$ 4x^2 + x - 5 $$ **Domain:** All real numbers. 2. **$$ f - g $$:** $$ -4x^2 + x - 5 $$ **Domain:** All real numbers. 3. **$$ f \cdot g $$:** $$ 4x^3 - 20x^2 $$ **Domain:** All real numbers. 4. **$$ \frac{f}{g} $$:** $$ \frac{x - 5}{4x^2} $$ **Domain:** All real numbers except $$ x = 0 $$.

Find , and . Determine the domain for each function.

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