Which of the following real valued functions is/are not even functions?

Options

A. `f(x)=x^(3) sin x`
B. `f(x)=x^(2) cos x`
C. `f(x)=e^(x) x^(3) sin x`
D. `f(x)=x-[x]` where [x] denote the greatest integer less than or equal to x

Answer

To determine which of the given real-valued functions are not even functions, we need to recall the definition of an even function. A function \( f(x) \) is considered even if it satisfies the condition:

\[ f(-x) = f(x) \quad \text{for all } x \]

If this condition does not hold, then the function is not even. Let's analyze each function step by step.

### Step 1: Analyze the first function \( f(x) = x^3 \sin x \)

1. Calculate \( f(-x) \):
\[ f(-x) = (-x)^3 \sin(-x) = -x^3 (-\sin x) = x^3 \sin x \]
2. Compare \( f(-x) \) with \( f(x) \):
\[ f(-x) = x^3 \sin x = f(x) \]
3. Conclusion: Since \( f(-x) = f(x) \), this function is an even function.

### Step 2: Analyze the second function \( f(x) = x^2 \cos x \)

1. Calculate \( f(-x) \):
\[ f(-x) = (-x)^2 \cos(-x) = x^2 \cos x \]
2. Compare \( f(-x) \) with \( f(x) \):
\[ f(-x) = x^2 \cos x = f(x) \]
3. Conclusion: Since \( f(-x) = f(x) \), this function is also an even function.

### Step 3: Analyze the third function \( f(x) = e^x \)

1. Calculate \( f(-x) \):
\[ f(-x) = e^{-x} \]
2. Compare \( f(-x) \) with \( f(x) \):
\[ f(-x) = e^{-x} \neq e^x = f(x) \]
3. Conclusion: Since \( f(-x) \neq f(x) \), this function is not an even function.

### Step 4: Analyze the fourth function \( f(x) = x - \lfloor x \rfloor \)

1. Recognize that \( f(x) \) represents the fractional part of \( x \).
2. Calculate \( f(-x) \):
\[ f(-x) = -x - \lfloor -x \rfloor \]
The floor function \( \lfloor -x \rfloor \) is not equal to \( -\lfloor x \rfloor \) for non-integer values of \( x \).
3. For example, if \( x = 1.02 \):
\[ f(1.02) = 1.02 - 1 = 0.02 \]
\[ f(-1.02) = -1.02 - (-2) = 0.98 \]
4. Compare \( f(-x) \) with \( f(x) \):
\[ f(-x) \neq f(x) \]
5. Conclusion: Since \( f(-x) \neq f(x) \), this function is not an even function.

### Final Conclusion

The functions that are **not even functions** are:
- \( f(x) = e^x \)
- \( f(x) = x - \lfloor x \rfloor \)

### Summary of Results
- **Not Even Functions**:
- \( f(x) = e^x \)
- \( f(x) = x - \lfloor x \rfloor \)