(a) $$ (f+g)(x)=x^{2}-15 x+38 $$ (b) $$ (f-g)(x)=x^{2}-9 x+34 $$ (c) $$ (f g)(x)=x^{2} $$ (d) $$ \left(\frac{f}{g}\right)(x)=\frac{(x-6)^{2}}{2-3 x} $$ Question Help: Dideo

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Hello! It appears you've provided a set of equations involving two functions, $$ f(x) $$ and $$ g(x) $$, labeled from (a) to (d). You might be looking to determine the explicit forms of $$ f(x) $$ and $$ g(x) $$ based on these relationships. Let's analyze each part step-by-step to find $$ f(x) $$ and $$ g(x) $$.

### Given:

1. **(a) Sum of Functions:**
   $$
   (f + g)(x) = x^{2} - 15x + 38
   $$
   
2. **(b) Difference of Functions:**
   $$
   (f - g)(x) = x^{2} - 9x + 34
   $$
   
3. **(c) Product of Functions:**
   $$
   (f \cdot g)(x) = x^{2}
   $$
   
4. **(d) Quotient of Functions:**
   $$
   \left( \frac{f}{g} \right)(x) = \frac{(x - 6)^{2}}{2 - 3x}
   $$

### Step 1: Determine $$ f(x) $$ and $$ g(x) $$ Using (a) and (b)

First, we'll use equations (a) and (b) to solve for $$ f(x) $$ and $$ g(x) $$.

**a.** Sum the equations to eliminate $$ g(x) $$:

$$
(f + g)(x) + (f - g)(x) = (x^{2} - 15x + 38) + (x^{2} - 9x + 34)
$$

$$
2f(x) = 2x^{2} - 24x + 72
$$

$$
f(x) = x^{2} - 12x + 36
$$

**b.** Subtract equation (b) from equation (a) to eliminate $$ f(x) $$:

$$
(f + g)(x) - (f - g)(x) = (x^{2} - 15x + 38) - (x^{2} - 9x + 34)
$$

$$
2g(x) = -6x + 4
$$

$$
g(x) = -3x + 2
$$

So far, we have:
$$
f(x) = x^{2} - 12x + 36 = (x - 6)^2
$$
$$
g(x) = -3x + 2
$$

### Step 2: Verify with Equations (c) and (d)

Now, let's check if these functions satisfy the remaining equations.

**c.** Product of Functions:
$$
f(x) \cdot g(x) = (x^{2} - 12x + 36)(-3x + 2)
$$
$$
= -3x^3 + 2x^{2} + 36x - 24x^{2} + 72x - 72
$$
$$
= -3x^3 - 22x^{2} + 108x - 72
$$
This does **not** simplify to $$ x^{2} $$, so equation (c) is **not** satisfied by the current $$ f(x) $$ and $$ g(x) $$.

**d.** Quotient of Functions:
$$
\frac{f(x)}{g(x)} = \frac{(x^{2} - 12x + 36)}{(-3x + 2)} = \frac{(x - 6)^2}{2 - 3x}
$$
This matches the given expression in equation (d), so **equation (d) is satisfied**.

### Step 3: Analyzing Consistency

The functions derived from equations (a) and (b) satisfy both the sum, difference, and quotient but **do not satisfy the product** as given in equation (c). This inconsistency suggests one of the following:

1. **Independent Problems:** Each part (a) to (d) might be independent problems where you're asked to find $$ f(x) $$ and $$ g(x) $$ based solely on each individual equation. In this case, you'd solve each part separately.

2. **Typographical Error:** There might be a typo in one of the equations, especially since the product from the derived functions doesn't align with part (c).

3. **Additional Constraints Needed:** To satisfy all four equations simultaneously, additional constraints or corrections may be necessary.

### Step 4: Next Steps

- **Clarify the Problem:** If all four parts are meant to hold simultaneously, double-check the given equations for any possible errors.

- **Solve Independently:** If each part is a separate problem, solve for $$ f(x) $$ and $$ g(x) $$ individually using the relevant equations.

- **Seek More Information:** Provide additional context or instructions if available, which might help in accurately addressing the problem.

---

**Summary of Derived Functions:**

Based on parts (a) and (b):

$$
f(x) = (x - 6)^2
$$
$$
g(x) = -3x + 2
$$

These satisfy parts (a), (b), and (d) but **not** part (c).

If you need further assistance or clarification on a specific part, feel free to ask!
Based on the given equations: - $$ f(x) = (x - 6)^2 $$ - $$ g(x) = -3x + 2 $$ These functions satisfy parts (a), (b), and (d) but do not satisfy part (c).

(a) \( (f+g)(x)=x^{2}-15 x+38 \) (b) \( (f-g)(x)=x^{2}-9 x+34 \) (c) \( (f g)(x)=x^{2} \) (d) \( \left(\frac{f}{g}\right)(x)=\frac{(x-6)^{2}}{2-3 x} \) Question Help: Dideo

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