The function `f (x)=4x^(2)-25and g (x) =-4x^(2) + 25` are graphed in the xy-plane above. The point where the two functions intersect are (z,0) and `(-z,0).` What is the value of z ?

Options

A. `0.5`
B. `1.0`
C. `2.5`
D. `4.0`

Explanation

To find the value of \( z \) where the functions \( f(x) = 4x^2 - 25 \) and \( g(x) = -4x^2 + 25 \) 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**:
\[ f(x) = g(x) \]
This gives us the equation:
\[ 4x^2 - 25 = -4x^2 + 25 \]

2. **Combine Like Terms**:
Move all terms involving \( x^2 \) to one side and constant terms to the other side:
\[ 4x^2 + 4x^2 = 25 + 25 \]
Simplifying this, we get:
\[ 8x^2 = 50 \]

3. **Divide by 8**:
To isolate \( x^2 \), divide both sides by 8:
\[ x^2 = \frac{50}{8} \]
Simplifying the fraction:
\[ x^2 = \frac{25}{4} \]

4. **Take the Square Root**:
To find \( x \), take the square root of both sides:
\[ x = \pm \sqrt{\frac{25}{4}} \]
This simplifies to:
\[ x = \pm \frac{5}{2} \]

5. **Identify the Value of \( z \)**:
Since the points of intersection are \( (z, 0) \) and \( (-z, 0) \), we take the positive value:
\[ z = \frac{5}{2} = 2.5 \]

### Final Answer:
The value of \( z \) is \( 2.5 \).