https://file.aiquickdraw.com/mathbot/file/1726059096709hedkeof9.jpg

The image shows the following two functions and questions:

### Given Functions:
1. \( f(x) = \frac{x^2 - 3x - 28}{x^2 - 8x + 16} \)
2. \( g(x) = \frac{x^2 - 5x + 41}{x^2 - 49} \)

### Questions:
1. Find \( (f \cdot g)(x) \).
2. Find \( \frac{f(x)}{g(x)} \).

### Let's go through each problem step by step:

#### 1. \( (f \cdot g)(x) \)
To multiply the two functions, you multiply their numerators and their denominators:

\[
(f \cdot g)(x) = \frac{(x^2 - 3x - 28)(x^2 - 5x + 41)}{(x^2 - 8x + 16)(x^2 - 49)}
\]

We will expand the expressions for the numerator and simplify the denominator using factoring where possible.

#### 2. \( \frac{f(x)}{g(x)} \)
To divide \( f(x) \) by \( g(x) \), you multiply \( f(x) \) by the reciprocal of \( g(x) \):

\[
\frac{f(x)}{g(x)} = \frac{\frac{x^2 - 3x - 28}{x^2 - 8x + 16}}{\frac{x^2 - 5x + 41}{x^2 - 49}} = \frac{(x^2 - 3x - 28)(x^2 - 49)}{(x^2 - 8x + 16)(x^2 - 5x + 41)}
\]

Again, we will simplify the expressions.

I will now proceed to compute and simplify both expressions.### Results:

1. \( (f \cdot g)(x) = \frac{x^3 - x^2 + 21x + 164}{x^3 - x^2 - 40x + 112} \)
   
2. \( \frac{f(x)}{g(x)} = \frac{x^4 - 3x^3 - 77x^2 + 147x + 1372}{x^4 - 13x^3 + 97x^2 - 408x + 656} \)

These are the simplified forms of the given problems.

Would you like further explanation on how these were derived, or do you have any additional questions?

Here are 5 related questions to expand on this concept:
1. How do you factor quadratic expressions to simplify these functions further?
2. What happens if there are common factors in the numerator and denominator?
3. How do you find the domain of \( f(x) \cdot g(x) \) and \( \frac{f(x)}{g(x)} \)?
4. Can you graph these functions to understand their behavior?
5. How would you compute the limits of \( f(x) \cdot g(x) \) as \( x \to \infty \)?

**Tip:** When multiplying or dividing rational functions, always check if you can simplify by factoring common terms in both the numerator and denominator.

Given functions f(x) = (x^2 - 3x - 28) / (x^2 - 8x + 16) and g(x) = (x^2 - 5x + 41) / (x^2 - 49), find (f * g)(x) and f(x) / g(x).

Solve the multiplication and division of the rational functions f(x) = (x^2 - 3x - 28) / (x^2 - 8x + 16) and g(x) = (x^2 - 5x + 41) / (x^2 - 49). Learn key algebraic concepts and techniques for simplifying rational expressions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation