Math Problem Statement

Suppose that the functions f and g are defined as follows: f(x) = x - 5 and g(x) = (x - 4)(x + 4). (a) Find (f/g)(-2). (b) Find all values that are NOT in the domain of f/g.

Solution

Let's go through the problem you posted.

We are given two functions:

  • f(x)=x5f(x) = x - 5
  • g(x)=(x4)(x+4)g(x) = (x - 4)(x + 4)

(a) Find (fg)(2)\left( \frac{f}{g} \right)(-2).

This means we need to evaluate f(x)g(x)\frac{f(x)}{g(x)} at x=2x = -2.

  1. First, find f(2)f(-2): f(2)=25=7f(-2) = -2 - 5 = -7

  2. Now, find g(2)g(-2): g(2)=(24)(2+4)=(6)(2)=12g(-2) = (-2 - 4)(-2 + 4) = (-6)(2) = -12

  3. Now, calculate (fg)(2)\left( \frac{f}{g} \right)(-2): (fg)(2)=712=712\left( \frac{f}{g} \right)(-2) = \frac{-7}{-12} = \frac{7}{12}

(b) Find all values that are NOT in the domain of fg\frac{f}{g}.

The domain of fg\frac{f}{g} consists of all values of xx for which g(x)0g(x) \neq 0. So, we need to find when g(x)=0g(x) = 0.

g(x)=(x4)(x+4)g(x) = (x - 4)(x + 4) equals zero when x=4x = 4 or x=4x = -4. Thus, the values that are not in the domain of fg\frac{f}{g} are x=4x = 4 and x=4x = -4.


Final Answers:

  • (a) (fg)(2)=712\left( \frac{f}{g} \right)(-2) = \frac{7}{12}
  • (b) Values not in the domain of fg\frac{f}{g}: 4,44, -4

Would you like further clarification or details on any part of this? Here are some related questions you might find useful:

  1. How do you determine the domain of a quotient of functions?
  2. What happens if you try to divide by zero in functions?
  3. How do you factor quadratics like g(x)=(x4)(x+4)g(x) = (x - 4)(x + 4)?
  4. What are the key differences between the domain of f(x)f(x) and g(x)g(x)?
  5. How can you visually represent the domain and range of a function?

Tip: Always remember to check for values that make the denominator zero when dealing with rational functions!

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Math Problem Analysis

Mathematical Concepts

Algebra
Functions
Domain and Range
Quotient of Functions

Formulas

f(x) = x - 5
g(x) = (x - 4)(x + 4)
(f/g)(x) = f(x) / g(x)

Theorems

Division by Zero Theorem (the denominator cannot be zero)
Zero Product Property (a product of two numbers is zero if at least one of the numbers is zero)

Suitable Grade Level

Grades 9-11