Question

F of x equals 2x squared- x - 6
Gof x equals x squared+ 3x - 10
What is f over g of X then find the domain of f over g of X

Answer 1

To find \( \frac{f}{g}(x) \), we first need to express \( f(x) \) and \( g(x) \):

- \( f(x) = 2x^2 - x - 6 \)
- \( g(x) = x^2 + 3x - 10 \)

### Step 1: Calculate \( \frac{f}{g}(x) \)

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

### Step 2: Factor \( f(x) \) and \( g(x) \)

**Factoring \( f(x) \):**

To factor \( 2x^2 - x - 6 \), we need two numbers that multiply to \( 2 \cdot (-6) = -12 \) and add to \(-1\). The numbers \(-4\) and \(3\) work.

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

**Factoring \( g(x) \):**

To factor \( x^2 + 3x - 10 \), we need two numbers that multiply to \(-10\) and add to \(3\). The numbers \(5\) and \(-2\) work.

\[
g(x) = x^2 + 5x - 2x - 10 = (x + 5)(x - 2)
\]

### Step 3: Rewrite \( \frac{f}{g}(x) \)

Using the factored forms:

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

We can simplify this fraction:

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

### Step 4: Determine the Domain of \( \frac{f}{g}(x) \)

The domain of \( \frac{f}{g}(x) \) is all \( x \) values for which \( g(x) \) is not zero.

**To find where \( g(x) = 0 \):**

Set \( g(x) = x^2 + 3x - 10 = 0 \).
Factoring gives:

\[
(x + 5)(x - 2) = 0
\]

This gives the roots:
- \( x + 5 = 0 \) ⇒ \( x = -5 \)
- \( x - 2 = 0 \) ⇒ \( x = 2 \)

Thus, \( g(x) \) is zero at \( x = -5 \) and \( x = 2 \).

**Therefore, the domain of \( \frac{f}{g}(x) \) is:**

\[
\text{Domain: } x \in \mathbb{R} \text{, where } x \neq -5 \text{ and } x \neq 2.
\]

In interval notation, the domain is:

\[
(-\infty, -5) \cup (-5, 2) \cup (2, \infty)
\]