Questions: Find all numbers that are not in the domain of the function. Then give the domain using set notation. f(x) = (2x+18) / (4x^2+2x-2) A. 2,-1 ; x x ≠ 2,-1 B. -9,-1, 1/2 ; x x ≠ -9,-1, 1/2 C. -1/2, 1 ; x x ≠ -1/2, 1 D. 1/2,-1 ; x x ≠ 1/2,-1

Find all numbers that are not in the domain of the function. Then give the domain using set notation.

f(x) = (2x+18) / (4x^2+2x-2)

A. 2,-1 ; x x ≠ 2,-1

B. -9,-1, 1/2 ; x x ≠ -9,-1, 1/2

C. -1/2, 1 ; x x ≠ -1/2, 1

D. 1/2,-1 ; x x ≠ 1/2,-1

Transcript text: Find all numbers that are not in the domain of the function. Then give the domain using set notation. \[ f(x)=\frac{2 x+18}{4 x^{2}+2 x-2} \] A. $2,-1 ;\{x \mid x \neq 2,-1\}$ B. $-9,-1, \frac{1}{2} ;\left\{x \mid x \neq-9,-1, \frac{1}{2}\right\}$ C. $-\frac{1}{2}, 1 ;\left\{x \left\lvert\, x \neq-\frac{1}{2}\right., 1\right\}$ D. $\frac{1}{2},-1 ;\left\{x \left\lvert\, x \neq \frac{1}{2}\right.,-1\right\}$

Solution

Solution Steps

To find the numbers that are not in the domain of the function, we need to identify the values of $ x $ that make the denominator zero, as division by zero is undefined. We will solve the equation $ 4x^2 + 2x - 2 = 0 $ to find these values. Once we have these values, the domain of the function will be all real numbers except these values.

Step 1: Identify the Denominator

The function given is

\[ f(x) = \frac{2x + 18}{4x^2 + 2x - 2} \]

To find the domain, we need to determine when the denominator is equal to zero:

\[ 4x^2 + 2x - 2 = 0 \]

Step 2: Solve for Excluded Values

Using the quadratic formula, we find the roots of the equation. The solutions are:

\[ x = -1 \quad \text{and} \quad x = \frac{1}{2} \]

These values make the denominator zero, and thus they are excluded from the domain.

Step 3: State the Domain

The domain of the function can be expressed in set notation as:

\[ \{ x \mid x \neq -1, \, x \neq \frac{1}{2} \} \]