Which of the following functions are even, and which are odd :
(a) `f(x)=root3((1-x)^(2))+root3((1+x)^(2))`,
(b) `f(x)=x^(2)-|x|,`
(c) `f(x)=x sin^(2) x-x^(3),`
(d) `f(x)=(1+2^(x))^(2)//2^(x)?`

Answer

To determine whether the given functions are even or odd, we will use the definitions of even and odd functions:

- A function \( f(x) \) is **even** if \( f(-x) = f(x) \) for all \( x \).
- A function \( f(x) \) is **odd** if \( f(-x) = -f(x) \) for all \( x \).

Now, let's analyze each function step by step.

### (a) \( f(x) = \sqrt[3]{(1-x)^2} + \sqrt[3]{(1+x)^2} \)

1. **Calculate \( f(-x) \)**:
\[ f(-x) = \sqrt[3]{(1-(-x))^2} + \sqrt[3]{(1+(-x))^2} = \sqrt[3]{(1+x)^2} + \sqrt[3]{(1-x)^2} \]
2. **Compare \( f(-x) \) and \( f(x) \)**:
\[ f(-x) = \sqrt[3]{(1+x)^2} + \sqrt[3]{(1-x)^2} = f(x) \]
Thus, \( f(x) \) is **even**.

### (b) \( f(x) = x^2 - |x| \)

1. **Calculate \( f(-x) \)**:
\[ f(-x) = (-x)^2 - |-x| = x^2 - |x| \]
2. **Compare \( f(-x) \) and \( f(x) \)**:
\[ f(-x) = x^2 - |x| = f(x) \]
Thus, \( f(x) \) is **even**.

### (c) \( f(x) = x \sin^2 x - x^3 \)

1. **Calculate \( f(-x) \)**:
\[ f(-x) = (-x) \sin^2(-x) - (-x)^3 = -x \sin^2 x + x^3 \]
(Note: \( \sin(-x) = -\sin x \) implies \( \sin^2(-x) = \sin^2 x \))
2. **Compare \( f(-x) \) and \( f(x) \)**:
\[ f(-x) = -x \sin^2 x + x^3 = -\left(x \sin^2 x - x^3\right) = -f(x) \]
Thus, \( f(x) \) is **odd**.

### (d) \( f(x) = \frac{(1 + 2^x)^2}{2^x} \)

1. **Calculate \( f(-x) \)**:
\[ f(-x) = \frac{(1 + 2^{-x})^2}{2^{-x}} = (1 + 2^{-x})^2 \cdot 2^x \]
Expanding \( (1 + 2^{-x})^2 \):
\[ = (1 + 2^{-x})^2 \cdot 2^x = (1 + 2^{-x})^2 \cdot 2^x = (1 + 2^{-x})^2 \cdot 2^x \]
This simplifies to:
\[ = \frac{(2^x + 1)^2}{2^x} = f(x) \]
2. **Compare \( f(-x) \) and \( f(x) \)**:
\[ f(-x) = f(x) \]
Thus, \( f(x) \) is **even**.

### Summary of Results:
- (a) Even
- (b) Even
- (c) Odd
- (d) Even