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

We are given two functions:

- \( f(x) = x - 8 \)
- \( g(x) = 9x^2 \)

Let's go through parts (a)-(h) systematically:

---

### (a) \( (f+g)(x) \) and its domain:

#### Step 1: Find \( (f+g)(x) \)
\[
(f+g)(x) = f(x) + g(x) = (x - 8) + (9x^2)
\]
\[
(f+g)(x) = 9x^2 + x - 8
\]

#### Step 2: Find the domain
Since both \( f(x) = x - 8 \) and \( g(x) = 9x^2 \) are polynomials, they are defined for all real values of \( x \). Therefore, the domain is:

\[
\text{Domain of } (f+g)(x): (-\infty, \infty)
\]

---

### (b) \( (f-g)(x) \) and its domain:

#### Step 1: Find \( (f-g)(x) \)
\[
(f-g)(x) = f(x) - g(x) = (x - 8) - (9x^2)
\]
\[
(f-g)(x) = -9x^2 + x - 8
\]

#### Step 2: Find the domain
Again, since \( f(x) \) and \( g(x) \) are polynomials, their difference is also defined for all real numbers. So, the domain is:

\[
\text{Domain of } (f-g)(x): (-\infty, \infty)
\]

---

### (c) \( (fg)(x) \) and its domain:

#### Step 1: Find \( (fg)(x) \)
\[
(fg)(x) = f(x) \cdot g(x) = (x - 8) \cdot (9x^2)
\]
\[
(fg)(x) = 9x^2(x - 8) = 9x^3 - 72x^2
\]

#### Step 2: Find the domain
As both \( f(x) \) and \( g(x) \) are polynomials, their product is also a polynomial, which is defined for all real numbers. So, the domain is:

\[
\text{Domain of } (fg)(x): (-\infty, \infty)
\]

---

### (d) \( \left(\frac{f}{g}\right)(x) \) and its domain:

#### Step 1: Find \( \left(\frac{f}{g}\right)(x) \)
\[
\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{x - 8}{9x^2}
\]

#### Step 2: Find the domain
For the quotient \( \frac{f(x)}{g(x)} \), we must exclude any \( x \) that makes \( g(x) = 9x^2 = 0 \). Solving for \( x \):

\[
9x^2 = 0 \quad \Rightarrow \quad x = 0
\]

Thus, the domain excludes \( x = 0 \). The domain is:

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

---

### (e) \( f(3) \)

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

---

### (f) \( g(3) \)

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

---

### (g) \( (f+g)(3) \)

Substitute \( x = 3 \) into \( (f+g)(x) = 9x^2 + x - 8 \):
\[
(f+g)(3) = 9(3)^2 + 3 - 8 = 9 \times 9 + 3 - 8 = 81 + 3 - 8 = 76
\]

---

### (h) \( \left(\frac{f}{g}\right)(3) \)

Substitute \( x = 3 \) into \( \left(\frac{f}{g}\right)(x) = \frac{x - 8}{9x^2} \):
\[
\left(\frac{f}{g}\right)(3) = \frac{3 - 8}{9(3)^2} = \frac{-5}{9 \times 9} = \frac{-5}{81}
\]

---

### Summary:

- (a) \( (f+g)(x) = 9x^2 + x - 8 \), domain: \( (-\infty, \infty) \)
- (b) \( (f-g)(x) = -9x^2 + x - 8 \), domain: \( (-\infty, \infty) \)
- (c) \( (fg)(x) = 9x^3 - 72x^2 \), domain: \( (-\infty, \infty) \)
- (d) \( \left(\frac{f}{g}\right)(x) = \frac{x - 8}{9x^2} \), domain: \( (-\infty, 0) \cup (0, \infty) \)
- (e) \( f(3) = -5 \)
- (f) \( g(3) = 81 \)
- (g) \( (f+g)(3) = 76 \)
- (h) \( \left(\frac{f}{g}\right)(3) = \frac{-5}{81} \)

Would you like further explanations on any part?

### Related questions:
1. What is the composition of two functions, such as \( (f \circ g)(x) \)?
2. How do you find the inverse of a function like \( f(x) = x - 8 \)?
3. What are the critical points of \( f(x) \) and \( g(x) \)?
4. How do you graph polynomial functions like \( f(x) \) and \( g(x) \)?
5. How would you find asymptotes for rational functions like \( \frac{f(x)}{g(x)} \)?

### Tip: 
For rational functions, always check for points where the denominator equals zero to determine the domain.

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

Explore the algebraic operations and domain of the functions f(x) = x - 8 and g(x) = 9x^2. Learn how to add, subtract, multiply, and divide these functions while determining the domain. Suitable for high school students, Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation