Question

What is (f/g) (x) Also, please find the domain of (f/g) (x)
f(x)= 2x^2-x-6
g(x) = x^2+3x-10

Answer 1

To find \((f/g)(x)\), we first need to compute the expression:

\[
\frac{f(x)}{g(x)} = \frac{2x^2 - x - 6}{x^2 + 3x - 10}
\]

Next, we will factor both \(f(x)\) and \(g(x)\) to simplify the expression.

### Step 1: Factor \(f(x) = 2x^2 - x - 6\)

To factor \(f(x)\), we look for two numbers that multiply to \(2 \cdot (-6) = -12\) and add up to \(-1\). These numbers are \(3\) and \(-4\). Thus, we can rewrite \(f(x)\):

\[
f(x) = 2x^2 - 4x + 3x - 6 = 2x(x - 2) + 3(x - 2) = (2x + 3)(x - 2)
\]

### Step 2: Factor \(g(x) = x^2 + 3x - 10\)

To factor \(g(x)\), we need two numbers that multiply to \(-10\) and add to \(3\). These numbers are \(5\) and \(-2\). Thus,

\[
g(x) = (x + 5)(x - 2)
\]

### Step 3: Simplify \((f/g)(x)\)

Now we can rewrite the expression for \((f/g)(x)\):

\[
\frac{f(x)}{g(x)} = \frac{(2x + 3)(x - 2)}{(x + 5)(x - 2)}
\]

Notice that \(x - 2\) is a common factor. We can simplify it, but we need to keep in mind that \(x\) cannot be equal to \(2\) because it would make the denominator zero:

\[
(f/g)(x) = \frac{2x + 3}{x + 5}, \quad x \neq 2
\]

### Step 4: Find the domain of \((f/g)(x)\)

The domain of \((f/g)(x)\) consists of all real numbers except where the denominator is zero. The denominator of \((f/g)(x)\) is \(g(x) = (x + 5)(x - 2)\).

Setting \(g(x)\) to zero to find the restrictions:

\[
x + 5 = 0 \implies x = -5
\]
\[
x - 2 = 0 \implies x = 2
\]

Therefore, the function \((f/g)(x)\) is undefined at \(x = -5\) and \(x = 2\).

### Conclusion

The domain of \((f/g)(x)\) is:

\[
\text{Domain: } (-\infty, -5) \cup (-5, 2) \cup (2, \infty)
\]

So, we can summarize:

\[
(f/g)(x) = \frac{2x + 3}{x + 5} \quad \text{for } x \neq -5, 2
\]

And the domain is \( (-\infty, -5) \cup (-5, 2) \cup (2, \infty) \).