Let  f(x)= x-5 and g(x)=x^2-1be 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

To express the function \( r(x) = \frac{x - 5}{2x^2 - 2} \) in terms of \( f(x) \) and \( g(x) \), where \( f(x) = x - 5 \) and \( g(x) = x^2 - 1 \), follow these steps:

1. **Simplify the denominator of \( r(x) \):**

   Notice that the denominator \( 2x^2 - 2 \) can be factored as:
   \[
   2x^2 - 2 = 2(x^2 - 1)
   \]

2. **Rewrite \( r(x) \) using the factored denominator:**

   Substitute this into \( r(x) \):
   \[
   r(x) = \frac{x - 5}{2(x^2 - 1)}
   \]

3. **Express \( r(x) \) in terms of \( f(x) \) and \( g(x) \):**

   Now substitute \( f(x) = x - 5 \) and \( g(x) = x^2 - 1 \):
   \[
   r(x) = \frac{f(x)}{2g(x)}
   \]

So, the function \( r(x) \) can be expressed as:
\[
r(x) = \frac{f(x)}{2g(x)}
\]

This means that \( r(x) \) is the quotient of \( f(x) \) and \( 2g(x) \).

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Would you like further details or have any questions?

Here are some related questions for practice:

1. What is the sum of \( f(x) \) and \( g(x) \)?
2. What is the product of \( f(x) \) and \( g(x) \)?
3. How would you express \( r(x) = \frac{x - 3}{x^2 + 1} \) using different functions?
4. Can you find the difference \( f(x) - g(x) \)?
5. How would the function \( h(x) = 3f(x) + 4g(x) \) be expressed?

**Tip:** When working with functions, always check if the expressions can be simplified or factored before substituting in known functions. This can make the process easier and more straightforward.

Learn how to express the quotient of functions f(x) = x - 5 and g(x) = x^2 - 1 as r(x) = (x - 5) / (2(x^2 - 1)). This problem involves simplifying and factoring the denominator, suitable for high school students studying functions and algebra.

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