`g (x) (2)/(2x ^(3) -12 x ^(2) -14 x )`
For Which of the following values of x is the function g (x) dewfined ?

Options

A. `-1`
B. 0
C. 1
D. 7

Answer

To determine for which values of \( x \) the function \( g(x) = \frac{2}{2x^3 - 12x^2 - 14x} \) is defined, we need to find when the denominator is not equal to zero. Here’s a step-by-step solution:

### Step 1: Set the Denominator Not Equal to Zero
We start with the expression in the denominator:
\[ 2x^3 - 12x^2 - 14x \neq 0 \]

### Step 2: Factor Out Common Terms
Notice that we can factor out \( 2x \) from the expression:
\[ 2x(x^2 - 6x - 7) \neq 0 \]

### Step 3: Set Each Factor Not Equal to Zero
For the product to be non-zero, each factor must be non-zero:
1. \( 2x \neq 0 \)
2. \( x^2 - 6x - 7 \neq 0 \)

From the first factor:
\[ 2x \neq 0 \implies x \neq 0 \]

### Step 4: Solve the Quadratic Equation
Now, we need to solve the quadratic equation \( x^2 - 6x - 7 = 0 \) to find the values of \( x \) that make the denominator zero. We can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = -6, c = -7 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-6)^2 - 4(1)(-7) = 36 + 28 = 64 \]

Now substituting into the quadratic formula:
\[ x = \frac{6 \pm \sqrt{64}}{2} = \frac{6 \pm 8}{2} \]

Calculating the two possible values:
1. \( x = \frac{14}{2} = 7 \)
2. \( x = \frac{-2}{2} = -1 \)

### Step 5: List All Restrictions
From our calculations, we have:
- \( x \neq 0 \)
- \( x \neq 7 \)
- \( x \neq -1 \)

### Conclusion
Thus, the function \( g(x) \) is defined for all values of \( x \) except \( 0, 7, \) and \( -1 \).

### Final Answer
The function \( g(x) \) is defined for \( x \) values other than \( 0, 7, \) and \( -1 \).

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