Find whether the function are one–one or many–one
`f(x) = (3x^(2))/(4pi) - cos pi x `

Answer

To determine whether the function \( f(x) = \frac{3x^2}{4\pi} - \cos(\pi x) \) is one-one or many-one, we will analyze its derivative and check for monotonicity.

### Step 1: Find the derivative of the function
The first step is to differentiate the function \( f(x) \).

\[ f'(x) = \frac{d}{dx}\left(\frac{3x^2}{4\pi}\right) - \frac{d}{dx}(\cos(\pi x)) \]

Using the power rule and the chain rule, we get:

\[ f'(x) = \frac{3 \cdot 2x}{4\pi} - (-\sin(\pi x) \cdot \pi) \]

This simplifies to:

\[ f'(x) = \frac{6x}{4\pi} + \pi \sin(\pi x) \]

Further simplifying gives:

\[ f'(x) = \frac{3x}{2\pi} + \pi \sin(\pi x) \]

### Step 2: Analyze the sign of the derivative
To determine if the function is monotonic (either always increasing or always decreasing), we need to analyze the sign of \( f'(x) \).

We need to check if \( f'(x) \) can be positive, negative, or zero for different values of \( x \):

\[ f'(x) = \frac{3x}{2\pi} + \pi \sin(\pi x) \]

### Step 3: Set the derivative to zero
To find critical points, we set the derivative equal to zero:

\[ \frac{3x}{2\pi} + \pi \sin(\pi x) = 0 \]

This can be rearranged to:

\[ \frac{3x}{2\pi} = -\pi \sin(\pi x) \]

### Step 4: Analyze the critical points
The equation \( \frac{3x}{2\pi} = -\pi \sin(\pi x) \) indicates that there could be multiple solutions for \( x \) where the function changes from increasing to decreasing or vice versa.

### Step 5: Conclusion about monotonicity
Since \( \sin(\pi x) \) oscillates between -1 and 1, the term \( \pi \sin(\pi x) \) will also oscillate. This means that \( f'(x) \) can be both positive and negative for different values of \( x \), indicating that the function is not monotonic.

### Final Result
Since the function \( f(x) \) is not monotonic, it is classified as a many-one function.