For the given functions f and​ g, complete parts​ (a)-(h). For parts​ (a)-(d), also find the domain.
f left parenthesis x right parenthesis equals x minus 8​; g left parenthesis x right parenthesis equals 9 x squared

Let's begin with the functions:

\[
f(x) = x - 8 \quad \text{and} \quad g(x) = 9x^2
\]

We'll go through parts (a)-(h) step by step:

### Part (a): \( (f+g)(x) \)
The sum of two functions is defined as:
\[
(f+g)(x) = f(x) + g(x)
\]
Substitute the given functions:
\[
(f+g)(x) = (x - 8) + 9x^2 = 9x^2 + x - 8
\]

**Domain of \( (f+g)(x) \):**

Since both functions \( f(x) \) and \( g(x) \) are defined for all real numbers, their sum is also defined for all real numbers. So, the domain of \( (f+g)(x) \) is:
\[
\text{Domain: } (-\infty, \infty)
\]

### Part (b): \( (f-g)(x) \)
The difference of two functions is defined as:
\[
(f-g)(x) = f(x) - g(x)
\]
Substitute the given functions:
\[
(f-g)(x) = (x - 8) - 9x^2 = -9x^2 + x - 8
\]

**Domain of \( (f-g)(x) \):**

Since both functions are defined for all real numbers, their difference is also defined for all real numbers. So, the domain of \( (f-g)(x) \) is:
\[
\text{Domain: } (-\infty, \infty)
\]

### Part (c): \( (fg)(x) \)
The product of two functions is defined as:
\[
(fg)(x) = f(x) \cdot g(x)
\]
Substitute the given functions:
\[
(fg)(x) = (x - 8)(9x^2) = 9x^3 - 72x^2
\]

**Domain of \( (fg)(x) \):**

The product is defined for all real numbers. So, the domain of \( (fg)(x) \) is:
\[
\text{Domain: } (-\infty, \infty)
\]

### Part (d): \( \left(\frac{f}{g}\right)(x) \)
The quotient of two functions is defined as:
\[
\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}
\]
Substitute the given functions:
\[
\left(\frac{f}{g}\right)(x) = \frac{x - 8}{9x^2}
\]

**Domain of \( \left(\frac{f}{g}\right)(x) \):**

For this quotient to be defined, \( g(x) \neq 0 \). Set \( g(x) = 9x^2 = 0 \), which gives \( x = 0 \). Therefore, the function is undefined at \( x = 0 \).

So, the domain of \( \left(\frac{f}{g}\right)(x) \) is:
\[
\text{Domain: } (-\infty, 0) \cup (0, \infty)
\]

### Part (e): \( f(3) \)
Substitute \( x = 3 \) into \( f(x) \):
\[
f(3) = 3 - 8 = -5
\]

### Part (f): \( g(3) \)
Substitute \( x = 3 \) into \( g(x) \):
\[
g(3) = 9(3^2) = 9(9) = 81
\]

### Part (g): \( (f+g)(3) \)
Use the expression for \( (f+g)(x) \) and substitute \( x = 3 \):
\[
(f+g)(3) = 9(3^2) + 3 - 8 = 81 + 3 - 8 = 76
\]

### Part (h): \( \left(\frac{f}{g}\right)(3) \)
Use the expression for \( \left(\frac{f}{g}\right)(x) \) and substitute \( x = 3 \):
\[
\left(\frac{f}{g}\right)(3) = \frac{3 - 8}{9(3^2)} = \frac{-5}{81}
\]

\[
\boxed{\frac{-5}{81}}
\]

---

Let me know if you'd like more details or explanations. Here are some related questions for further exploration:

1. What is the inverse of \( f(x) = x - 8 \)?
2. What is the range of \( g(x) = 9x^2 \)?
3. How would you find the intersection points of \( f(x) \) and \( g(x) \)?
4. What is the derivative of \( (f+g)(x) \)?
5. How do you find the domain of more complex composite functions like \( f(g(x)) \)?

**Tip:** Always check for undefined points (such as where the denominator equals zero) when working with quotients of functions.

For the given functions f and g, complete parts (a)-(h). For parts (a)-(d), also find the domain. f(x) = x - 8; g(x) = 9x^2.

Solve function operations including addition, subtraction, multiplication, and division for f(x) = x - 8 and g(x) = 9x^2. Learn how to find the domain for each operation and calculate specific values. Suitable for students in Grades 9-12.

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