Let 

*f*(*x*) = 4*x* + 24

 and 

*g*(*x*) = *x*2 − 36.


Find the domain of the following functions and simplify their expressions. (Enter your answer for each domain using interval notation.)

(a)    

(*g* − *f*)(*x*)




domain
 
 
(*g* − *f*)(*x*) =

x2−4x−60
 
 


(b)    

*f*

*g*

(*x*)




domain
 
*f*

*g*

(*x*) =

Let's start by analyzing each function and finding the domain for each operation.

### Given Functions:
\[
f(x) = 4x + 24
\]
\[
g(x) = x^2 - 36
\]

### (a) **Subtraction of Functions**: \( (g - f)(x) \)

The subtraction of the functions is given by:
\[
(g - f)(x) = g(x) - f(x)
\]

Substituting the given functions:
\[
(g - f)(x) = (x^2 - 36) - (4x + 24)
\]
Simplify the expression:
\[
(g - f)(x) = x^2 - 4x - 36 - 24 = x^2 - 4x - 60
\]

#### Domain of \( (g - f)(x) \):
- Both \( f(x) \) and \( g(x) \) are polynomials, which are defined for all real numbers.
- Therefore, the domain of \( (g - f)(x) \) is all real numbers.

**Domain**: \( (-\infty, \infty) \)

**Simplified Expression**: 
\[
(g - f)(x) = x^2 - 4x - 60
\]

### (b) **Division of Functions**: \( \frac{f(x)}{g(x)} \)

The division of the functions is given by:
\[
\frac{f(x)}{g(x)} = \frac{4x + 24}{x^2 - 36}
\]

#### Simplification:
Factor the denominator:
\[
x^2 - 36 = (x - 6)(x + 6)
\]

Thus, the expression simplifies to:
\[
\frac{f(x)}{g(x)} = \frac{4x + 24}{(x - 6)(x + 6)}
\]
We can factor the numerator:
\[
\frac{4(x + 6)}{(x - 6)(x + 6)}
\]
Cancel the common factor \( x + 6 \) (except when \( x = -6 \)):
\[
\frac{f(x)}{g(x)} = \frac{4}{x - 6} \quad \text{for } x \neq -6
\]

#### Domain of \( \frac{f(x)}{g(x)} \):
- The denominator \( g(x) = x^2 - 36 \) must not be zero, so \( x^2 - 36 = 0 \) implies \( x = \pm 6 \).
- Therefore, \( x = 6 \) and \( x = -6 \) must be excluded from the domain.

**Domain**: \( (-\infty, -6) \cup (-6, 6) \cup (6, \infty) \)

**Simplified Expression**: 
\[
\frac{f(x)}{g(x)} = \frac{4}{x - 6} \quad \text{for } x \neq -6
\]

### Summary:
(a) \((g - f)(x) = x^2 - 4x - 60\)
- **Domain**: \( (-\infty, \infty) \)

(b) \( \frac{f(x)}{g(x)} = \frac{4}{x - 6} \)
- **Domain**: \( (-\infty, -6) \cup (-6, 6) \cup (6, \infty) \)

Would you like further details on any part of the process, or do you have any questions?

Here are some related questions for further exploration:
1. How do you find the roots of the quadratic equation \( x^2 - 4x - 60 \)?
2. What are vertical asymptotes, and how do they relate to the domain of rational functions?
3. How would the domain change if the numerator of \( \frac{f(x)}{g(x)} \) was also a polynomial that could be factored?
4. What is the graphical interpretation of the domain of \( \frac{f(x)}{g(x)} \)?
5. How does the process change when dealing with complex numbers?

**Tip:** Always check for common factors in both the numerator and denominator when simplifying rational expressions to avoid mistakes in determining the domain.

Explore the domain and simplification of functions g(x) - f(x) and f(x) / g(x) with detailed solutions. Understand how to find the domain of each function and simplify their expressions effectively.

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