Let f(x) be a real valued function such that the area of an equilateral triangle with two of its vertices at (0, 0) and (x, f(x)) is `(sqrt(3))/(4)` square units. Then
f(x) is given by

Options

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

Answer

To find the function \( f(x) \) given that the area of an equilateral triangle with vertices at \( (0, 0) \) and \( (x, f(x)) \) is \( \frac{\sqrt{3}}{4} \) square units, we can follow these steps:

### Step 1: Understand the Area of an Equilateral Triangle
The area \( A \) of an equilateral triangle with side length \( s \) is given by the formula:
\[ A = \frac{\sqrt{3}}{4} s^2 \]
In our case, we need to find the side length \( s \) of the triangle formed by the points \( (0, 0) \) and \( (x, f(x)) \).

### Step 2: Calculate the Side Length
The distance \( s \) between the points \( (0, 0) \) and \( (x, f(x)) \) can be calculated using the distance formula:
\[ s = \sqrt{(x - 0)^2 + (f(x) - 0)^2} = \sqrt{x^2 + f(x)^2} \]

### Step 3: Set Up the Area Equation
Since the area of the triangle is given as \( \frac{\sqrt{3}}{4} \), we can set up the equation:
\[ \frac{\sqrt{3}}{4} = \frac{\sqrt{3}}{4} s^2 \]
Substituting \( s \) into the equation gives:
\[ \frac{\sqrt{3}}{4} = \frac{\sqrt{3}}{4} (x^2 + f(x)^2) \]

### Step 4: Simplify the Equation
We can cancel \( \frac{\sqrt{3}}{4} \) from both sides (assuming \( \sqrt{3} \neq 0 \)):
\[ 1 = x^2 + f(x)^2 \]

### Step 5: Solve for \( f(x) \)
Rearranging the equation gives:
\[ f(x)^2 = 1 - x^2 \]
Taking the square root of both sides results in:
\[ f(x) = \sqrt{1 - x^2} \quad \text{or} \quad f(x) = -\sqrt{1 - x^2} \]

### Step 6: Identify the Correct Option
From the options provided, the function \( f(x) \) can be expressed as:
\[ f(x) = \pm \sqrt{1 - x^2} \]
Thus, the correct answer is:
\[ f(x) = \sqrt{1 - x^2} \quad \text{(Option 3)} \]