Math Problem Statement

Let f(x)= x-5 and g(x)= x^2-1 be defined as in the previous example. Express the following functions as the sum, difference, quotient, or product of f and g of r(x)=x-5\2x^2-2

Solution

To express the function r(x)=x52x22r(x) = \frac{x-5}{2x^2-2} as a combination of f(x)f(x) and g(x)g(x), we start by analyzing how r(x)r(x) can be rewritten in terms of f(x)=x5f(x) = x - 5 and g(x)=x21g(x) = x^2 - 1.

Step 1: Factor the denominator of r(x)r(x)

First, we factor the denominator 2x222x^2 - 2: 2x22=2(x21)2x^2 - 2 = 2(x^2 - 1) Thus, we can rewrite r(x)r(x) as: r(x)=x52(x21)r(x) = \frac{x - 5}{2(x^2 - 1)}

Step 2: Express r(x)r(x) in terms of f(x)f(x) and g(x)g(x)

Since g(x)=x21g(x) = x^2 - 1, we can substitute g(x)g(x) into the expression: r(x)=f(x)2g(x)r(x) = \frac{f(x)}{2g(x)}

Final Answer:

The function r(x)=x52x22r(x) = \frac{x-5}{2x^2-2} can be expressed as: r(x)=f(x)2g(x)r(x) = \frac{f(x)}{2g(x)} This represents the quotient of f(x)f(x) and 2g(x)2g(x).

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Related Questions:

  1. How do you express the sum of f(x)f(x) and ( g(x

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Math Problem Analysis

Mathematical Concepts

Functions
Quotient of Functions
Factorization

Formulas

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Theorems

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Suitable Grade Level

High School