If g (x) be a function defined on [-1,1] . If the area of the equilateral triangle with two of its vertices at (0,0) and [x,g (x)]` is ` `sqrt(3 //4)` them the function is

Options

A. `g(x) = +- sqrt""(1-x^2)`
B. `g(x) = - sqrt""(1-x^2)`
C. `g(x) = sqrt""(1-x^2)`
D. `g(x) = +- sqrt""(1+x^2)`

Answer

To solve the problem step by step, we will find the function \( g(x) \) based on the given conditions.

### Step 1: Understand the Area of an Equilateral Triangle
The area \( A \) of an equilateral triangle with side length \( a \) is given by the formula:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
Given that the area is \( \frac{\sqrt{3}}{4} \), we can set up the equation:
\[ \frac{\sqrt{3}}{4} a^2 = \frac{\sqrt{3}}{4} \]
This implies:
\[ a^2 = 1 \quad \Rightarrow \quad a = 1 \quad \text{(since side length is positive)} \]

### Step 2: Find the Length of the Side \( a \)
The side length \( a \) of the triangle can also be determined using the distance formula between the points \( (0,0) \) and \( (x, g(x)) \):
\[ a = \sqrt{(x - 0)^2 + (g(x) - 0)^2} = \sqrt{x^2 + (g(x))^2} \]
Setting this equal to 1 (from Step 1):
\[ \sqrt{x^2 + (g(x))^2} = 1 \]

### Step 3: Square Both Sides
To eliminate the square root, we square both sides:
\[ x^2 + (g(x))^2 = 1 \]

### Step 4: Rearrange the Equation
Rearranging gives us:
\[ (g(x))^2 = 1 - x^2 \]

### Step 5: Solve for \( g(x) \)
Taking the square root of both sides, we find:
\[ g(x) = \pm \sqrt{1 - x^2} \]

### Conclusion
Thus, the function \( g(x) \) can be expressed as:
\[ g(x) = \sqrt{1 - x^2} \quad \text{or} \quad g(x) = -\sqrt{1 - x^2} \]
Since the problem does not specify which branch to use, both are valid solutions.