Find $$ f+g, f-g, f g $$ and $$ \frac{f}{g} $$. Determine the domain for each function. $$ f(x)=4 x-5, g(x)=x-7 $$ $$ (f+g)(x)=\square $$ (Simplify your answer.) What is the domain of $$ f+g $$ ? A. The domain of $$ f+g $$ is $$ \{\square $$. (Use a comma to separate answers as needed.) B. The domain of $$ f+g $$ is $$ \square $$. (Type your answer in interval notation.) ( $$ f-g)(x)=\square $$ (Simplify your answer.) What is the domain of $$ f-g $$ ? $$ f+g $$ is $$ \varnothing $$. A. The domain of $$ f-g $$ is $$ \} $$. (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 $$ f-g $$ is $$ \varnothing $$.

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UpStudy AI · Accepted Answer

To find $$ f+g $$, $$ f-g $$, $$ fg $$, and $$ \frac{f}{g} $$ for the functions $$ f(x) = 4x - 5 $$ and $$ g(x) = x - 7 $$, we will perform the calculations step by step.

### Step 1: Calculate $$ f+g $$

$$
(f+g)(x) = f(x) + g(x) = (4x - 5) + (x - 7)
$$

Now, simplify:

$$
(f+g)(x) = 4x - 5 + x - 7 = 5x - 12
$$

### Step 2: Determine the domain of $$ f+g $$

Since both $$ f(x) $$ and $$ g(x) $$ are linear functions, they are defined for all real numbers. Therefore, the domain of $$ f+g $$ is:

A. The domain of $$ f+g $$ is $$ \{\text{all real numbers}\} $$.

B. The domain of $$ f+g $$ is $$ (-\infty, \infty) $$.

### Step 3: Calculate $$ f-g $$

$$
(f-g)(x) = f(x) - g(x) = (4x - 5) - (x - 7)
$$

Now, simplify:

$$
(f-g)(x) = 4x - 5 - x + 7 = 3x + 2
$$

### Step 4: Determine the domain of $$ f-g $$

Similar to $$ f+g $$, since both $$ f(x) $$ and $$ g(x) $$ are defined for all real numbers, the domain of $$ f-g $$ is:

A. The domain of $$ f-g $$ is $$ \{\text{all real numbers}\} $$.

B. The domain of $$ f-g $$ is $$ (-\infty, \infty) $$.

### Step 5: Calculate $$ fg $$

$$
(fg)(x) = f(x) \cdot g(x) = (4x - 5)(x - 7)
$$

Now, expand:

$$
(fg)(x) = 4x^2 - 28x - 5x + 35 = 4x^2 - 33x + 35
$$

### Step 6: Determine the domain of $$ fg $$

Since both $$ f(x) $$ and $$ g(x) $$ are defined for all real numbers, the domain of $$ fg $$ is:

A. The domain of $$ fg $$ is $$ \{\text{all real numbers}\} $$.

B. The domain of $$ fg $$ is $$ (-\infty, \infty) $$.

### Step 7: Calculate $$ \frac{f}{g} $$

$$
\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{4x - 5}{x - 7}
$$

### Step 8: Determine the domain of $$ \frac{f}{g} $$

The function $$ \frac{f}{g} $$ is undefined when $$ g(x) = 0 $$:

$$
x - 7 = 0 \implies x = 7
$$

Thus, the domain of $$ \frac{f}{g} $$ is:

A. The domain of $$ \frac{f}{g} $$ is $$ \{\text{all real numbers except } 7\} $$.

B. The domain of $$ \frac{f}{g} $$ is $$ (-\infty, 7) \cup (7, \infty) $$.

### Summary of Results

1. $$ (f+g)(x) = 5x - 12 $$
   - Domain: $$ (-\infty, \infty) $$

2. $$ (f-g)(x) = 3x + 2 $$
   - Domain: $$ (-\infty, \infty) $$

3. $$ (fg)(x) = 4x^2 - 33x + 35 $$
   - Domain: $$ (-\infty, \infty) $$

4. $$ \left(\frac{f}{g}\right)(x) = \frac{4x - 5}{x - 7} $$
   - Domain: $$ (-\infty, 7) \cup (7, \infty) $$
$$ (f+g)(x) = 5x - 12 $$ Domain of $$ f+g $$: All real numbers, $$ (-\infty, \infty) $$ $$ (f-g)(x) = 3x + 2 $$ Domain of $$ f-g $$: All real numbers, $$ (-\infty, \infty) $$ $$ fg(x) = 4x^2 - 33x + 35 $$ Domain of $$ fg $$: All real numbers, $$ (-\infty, \infty) $$ $$ \frac{f}{g}(x) = \frac{4x - 5}{x - 7} $$ Domain of $$ \frac{f}{g} $$: All real numbers except $$ x = 7 $$, $$ (-\infty, 7) \cup (7, \infty) $$

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