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The function ` f ` and ` g ` , defined by ` f (x) = 8x ^ 2 - 2 ` and ` g (x) = -8 x ^ 2 + 2 `, are graphed in the `xy` - plane above. The graphs of ` f and g ` intersect at the points ` ( k, 0 ) and ( -k, 0) `. What is the value of `k ` ?

Options

A. ` ( 1 ) /(4) `
B. ` ( 1) /(2) `
C. ` 1 `
D. ` 2 `

Explanation

To find the value of \( k \) where the functions \( f(x) = 8x^2 - 2 \) and \( g(x) = -8x^2 + 2 \) intersect, we need to set the two functions equal to each other and solve for \( x \).

### Step-by-step Solution:

1. **Set the functions equal to each other:**
\[ f(x) = g(x) \]
This gives us:
\[ 8x^2 - 2 = -8x^2 + 2 \]

2. **Rearrange the equation:**
Add \( 8x^2 \) to both sides and add \( 2 \) to both sides:
\[ 8x^2 + 8x^2 = 2 + 2 \]
This simplifies to:
\[ 16x^2 = 4 \]

3. **Divide both sides by 16:**
\[ x^2 = \frac{4}{16} \]
Simplifying gives:
\[ x^2 = \frac{1}{4} \]

4. **Take the square root of both sides:**
\[ x = \pm \sqrt{\frac{1}{4}} = \pm \frac{1}{2} \]

5. **Identify the points of intersection:**
The points of intersection are \( \left( \frac{1}{2}, 0 \right) \) and \( \left( -\frac{1}{2}, 0 \right) \).

6. **Determine the value of \( k \):**
From the points of intersection, we can see that \( k = \frac{1}{2} \).

### Final Answer:
\[ k = \frac{1}{2} \]