Math Problem Statement

Simplify the expression f(x)/g(x), where f(x) = x^2 - 1 and g(x) = x + 1.

Solution

The mathematical expression provided consists of two functions f(x)f(x) and g(x)g(x), with the operation fg\frac{f}{g}. Here's the breakdown:

Given: f(x)=x21f(x) = x^2 - 1 g(x)=x+1g(x) = x + 1

Task: Simplify f(x)g(x)\frac{f(x)}{g(x)}

Step 1: Substitute f(x)f(x) and g(x)g(x):

f(x)g(x)=x21x+1\frac{f(x)}{g(x)} = \frac{x^2 - 1}{x + 1}

Step 2: Factorize the numerator:

The numerator x21x^2 - 1 is a difference of squares: x21=(x1)(x+1)x^2 - 1 = (x - 1)(x + 1)

So the expression becomes: f(x)g(x)=(x1)(x+1)x+1\frac{f(x)}{g(x)} = \frac{(x - 1)(x + 1)}{x + 1}

Step 3: Simplify:

Cancel out x+1x + 1 (for x1x \neq -1): f(x)g(x)=x1,for x1\frac{f(x)}{g(x)} = x - 1, \quad \text{for } x \neq -1

Final Answer:

f(x)g(x)=x1(valid for x1)\frac{f(x)}{g(x)} = x - 1 \quad \text{(valid for } x \neq -1 \text{)}


Would you like a more detailed explanation or additional examples? Let me know!

Here are 5 related questions to consider:

  1. What happens to the function at x=1x = -1 (undefined points)?
  2. How do you graph the simplified function f(x)g(x)\frac{f(x)}{g(x)}?
  3. What is the domain of f(x)g(x)\frac{f(x)}{g(x)}?
  4. How would you handle similar rational expressions with higher degrees?
  5. How do you check your work when simplifying rational expressions?

Tip: Always check the domain of a rational function before and after simplification, as certain values might make the denominator zero.

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Math Problem Analysis

Mathematical Concepts

Algebra
Rational Expressions
Factoring
Simplification

Formulas

Difference of Squares: a^2 - b^2 = (a - b)(a + b)

Theorems

Simplification of Rational Expressions

Suitable Grade Level

Grades 9-10