Math Problem Statement

Find f + g, f - g, fg, and f/g. Determine the domain for each function. f(x) = (7x + 2) / (x^2 - 36), g(x) = (6x - 4) / (x^2 - 36)

Solution

Let's break down and solve this problem step by step.

Given Functions:

f(x)=7x+2x236,g(x)=6x4x236f(x) = \frac{7x + 2}{x^2 - 36}, \quad g(x) = \frac{6x - 4}{x^2 - 36} We are asked to find the following:

  1. f(x)+g(x)f(x) + g(x)
  2. f(x)g(x)f(x) - g(x)
  3. f(x)g(x)f(x) \cdot g(x)
  4. f(x)g(x)\frac{f(x)}{g(x)}

We also need to determine the domain for each function.

Domain

The domain of a rational function is the set of all real numbers except those that make the denominator equal to zero. Both f(x)f(x) and g(x)g(x) have the denominator x236x^2 - 36, which can be factored as: x236=(x6)(x+6)x^2 - 36 = (x - 6)(x + 6) Thus, x=6x = 6 and x=6x = -6 are the values that make the denominator zero. Therefore, the domain of both f(x)f(x) and g(x)g(x) is: Domain of f(x),g(x):xR,x±6\text{Domain of } f(x), g(x): x \in \mathbb{R}, x \neq \pm 6

1. f(x)+g(x)f(x) + g(x)

Since f(x)f(x) and g(x)g(x) have the same denominator, we can directly add the numerators: f(x)+g(x)=7x+2x236+6x4x236=(7x+2)+(6x4)x236=13x2x236f(x) + g(x) = \frac{7x + 2}{x^2 - 36} + \frac{6x - 4}{x^2 - 36} = \frac{(7x + 2) + (6x - 4)}{x^2 - 36} = \frac{13x - 2}{x^2 - 36} The domain of f(x)+g(x)f(x) + g(x) is the same as that of f(x)f(x) and g(x)g(x): x±6x \neq \pm 6.

2. f(x)g(x)f(x) - g(x)

Similarly, subtracting the numerators: f(x)g(x)=7x+2x2366x4x236=(7x+2)(6x4)x236=x+6x236f(x) - g(x) = \frac{7x + 2}{x^2 - 36} - \frac{6x - 4}{x^2 - 36} = \frac{(7x + 2) - (6x - 4)}{x^2 - 36} = \frac{x + 6}{x^2 - 36} The domain of f(x)g(x)f(x) - g(x) is also x±6x \neq \pm 6.

3. f(x)g(x)f(x) \cdot g(x)

Multiplying the two functions: f(x)g(x)=7x+2x2366x4x236=(7x+2)(6x4)(x236)2f(x) \cdot g(x) = \frac{7x + 2}{x^2 - 36} \cdot \frac{6x - 4}{x^2 - 36} = \frac{(7x + 2)(6x - 4)}{(x^2 - 36)^2} Now, expanding the numerator: (7x+2)(6x4)=42x228x+12x8=42x216x8(7x + 2)(6x - 4) = 42x^2 - 28x + 12x - 8 = 42x^2 - 16x - 8 Thus: f(x)g(x)=42x216x8(x236)2f(x) \cdot g(x) = \frac{42x^2 - 16x - 8}{(x^2 - 36)^2} The domain remains x±6x \neq \pm 6.

4. f(x)g(x)\frac{f(x)}{g(x)}

Dividing f(x)f(x) by g(x)g(x): f(x)g(x)=7x+2x2366x4x236=7x+26x4\frac{f(x)}{g(x)} = \frac{\frac{7x + 2}{x^2 - 36}}{\frac{6x - 4}{x^2 - 36}} = \frac{7x + 2}{6x - 4} Here, we must exclude any values that make the denominator zero, i.e., 6x4=06x - 4 = 0, or x=23x = \frac{2}{3}. Therefore, the domain is: Domain of f(x)g(x):xR,x±6,x23\text{Domain of } \frac{f(x)}{g(x)}: x \in \mathbb{R}, x \neq \pm 6, x \neq \frac{2}{3}

Summary:

  1. f(x)+g(x)=13x2x236f(x) + g(x) = \frac{13x - 2}{x^2 - 36}, Domain: x±6x \neq \pm 6
  2. f(x)g(x)=x+6x236f(x) - g(x) = \frac{x + 6}{x^2 - 36}, Domain: x±6x \neq \pm 6
  3. f(x)g(x)=42x216x8(x236)2f(x) \cdot g(x) = \frac{42x^2 - 16x - 8}{(x^2 - 36)^2}, Domain: x±6x \neq \pm 6
  4. f(x)g(x)=7x+26x4\frac{f(x)}{g(x)} = \frac{7x + 2}{6x - 4}, Domain: x±6,x23x \neq \pm 6, x \neq \frac{2}{3}

Would you like more details or have any questions?

Here are some related questions for further understanding:

  1. How would the domain change if the denominators were different?
  2. How would the functions behave at the boundaries of their domain (near x=6x = 6 and x=6x = -6)?
  3. What would happen if the numerator of one function was a quadratic equation?
  4. Can you simplify complex rational expressions like f(x)g(x)f(x) \cdot g(x) further?
  5. How would adding more terms to the numerator affect the overall function?

Tip: Always simplify the function as much as possible before analyzing its domain to avoid mistakes with undefined values.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Rational Functions
Function Operations
Domain of Functions

Formulas

(f + g)(x) = f(x) + g(x)
(f - g)(x) = f(x) - g(x)
(f * g)(x) = f(x) * g(x)
(f / g)(x) = f(x) / g(x)

Theorems

Domain of a rational function is all real numbers except where the denominator equals zero

Suitable Grade Level

Grades 10-12