If f ( x ) = x 2 - x - 6 and g ( x ) = x - 3,

find f ( x ) / g ( x )

To find \(\frac{f(x)}{g(x)}\), we'll first substitute the given functions \(f(x)\) and \(g(x)\) and then simplify.

Given:
\[
f(x) = x^2 - x - 6
\]
\[
g(x) = x - 3
\]

Now, we want to find:
\[
\frac{f(x)}{g(x)} = \frac{x^2 - x - 6}{x - 3}
\]

To simplify this, we can try to factor the numerator \(x^2 - x - 6\).

The numerator \(x^2 - x - 6\) factors as:
\[
x^2 - x - 6 = (x - 3)(x + 2)
\]

So we have:
\[
\frac{x^2 - x - 6}{x - 3} = \frac{(x - 3)(x + 2)}{x - 3}
\]

As long as \(x \neq 3\), the \(x - 3\) terms cancel out, giving:
\[
\frac{f(x)}{g(x)} = x + 2
\]

**Final Answer:** 
\[
\frac{f(x)}{g(x)} = x + 2 \quad \text{for} \quad x \neq 3
\]

Would you like further details or have any questions about this solution?

Here are some related questions you might find interesting:
1. How do you factor a quadratic expression like \(x^2 - x - 6\)?
2. What happens if \(x = 3\) in the expression \(\frac{f(x)}{g(x)}\)?
3. Can you find the remainder when \(f(x)\) is divided by \(g(x)\)?
4. How does the graph of \(f(x)\) compare to the graph of \(g(x)\)?
5. What is the domain of the function \(\frac{f(x)}{g(x)}\)?

**Tip:** When simplifying expressions, always check if there are restrictions on the values of \(x\) due to potential division by zero.

Learn how to calculate \(\frac{f(x)}{g(x)}\) where \(f(x) = x^2 - x - 6\) and \(g(x) = x - 3\). This problem explores function operations in algebra, including the quotient of functions formula. Suitable for high school students studying algebra and function operations.

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