If `f(x)=x^(3) and g(x)=x^(2)//5`, then `f(x)-g(x)` will be

Options

A. Odd function
B. Even function
C. Neither (a) nor (b)
D. Both (a) and (b)

Answer

To solve the problem, we need to find the expression for \( f(x) - g(x) \) and determine whether this expression is an even function, an odd function, or neither.

### Step-by-step Solution:

1. **Define the Functions**:
We have two functions:
\[ f(x) = x^3 \]
\[ g(x) = \frac{x^2}{5} \]

2. **Find \( f(x) - g(x) \)**:
We need to compute:
\[ f(x) - g(x) = x^3 - \frac{x^2}{5} \]

3. **Check for Evenness or Oddness**:
To determine if the function \( f(x) - g(x) \) is even, odd, or neither, we will evaluate \( f(-x) - g(-x) \).

- Calculate \( f(-x) \):
\[ f(-x) = (-x)^3 = -x^3 \]

- Calculate \( g(-x) \):
\[ g(-x) = \frac{(-x)^2}{5} = \frac{x^2}{5} \]

- Now, compute \( f(-x) - g(-x) \):
\[ f(-x) - g(-x) = -x^3 - \frac{x^2}{5} \]

4. **Compare with \( -(f(x) - g(x)) \)**:
We need to check if:
\[ f(-x) - g(-x) = -(f(x) - g(x)) \]
First, calculate \( -(f(x) - g(x)) \):
\[ -(f(x) - g(x)) = -\left(x^3 - \frac{x^2}{5}\right) = -x^3 + \frac{x^2}{5} \]

5. **Final Comparison**:
Now we compare:
\[ f(-x) - g(-x) = -x^3 - \frac{x^2}{5} \]
with:
\[ -(f(x) - g(x)) = -x^3 + \frac{x^2}{5} \]

Since \( f(-x) - g(-x) \) is not equal to either \( f(x) - g(x) \) or \( -(f(x) - g(x)) \), we conclude that the function is neither even nor odd.

### Conclusion:
Thus, the final answer is that \( f(x) - g(x) \) is **neither an even function nor an odd function**.