If `f(x) = x^(2)` and g(x) = Ixl then find the values of:
(i) f+g, (ii) f-g, (iii) fg, (iv) 2f, (v) `f^(2)`, (vi) f+3

Answer

To solve the problem, we need to find various combinations of the functions \( f(x) = x^2 \) and \( g(x) = |x| \). Let's go through each part step by step.

### Step-by-Step Solution:

1. **Finding \( f + g \)**:
\[ f + g = f(x) + g(x) = x^2 + |x| \]
Thus,
\[ f + g = x^2 + |x| \]

2. **Finding \( f - g \)**:
\[ f - g = f(x) - g(x) = x^2 - |x| \]
Therefore,
\[ f - g = x^2 - |x| \]

3. **Finding \( fg \)**:
\[ fg = f(x) \cdot g(x) = x^2 \cdot |x| \]
This simplifies to:
\[ fg = x^2 |x| = x^3 \text{ (for } x \geq 0\text{) or } -x^3 \text{ (for } x < 0\text{)} \]

4. **Finding \( 2f \)**:
\[ 2f = 2 \cdot f(x) = 2 \cdot x^2 \]
Thus,
\[ 2f = 2x^2 \]

5. **Finding \( f^2 \)**:
\[ f^2 = (f(x))^2 = (x^2)^2 = x^4 \]
Therefore,
\[ f^2 = x^4 \]

6. **Finding \( f + 3 \)**:
\[ f + 3 = f(x) + 3 = x^2 + 3 \]
Thus,
\[ f + 3 = x^2 + 3 \]

### Summary of Results:
- (i) \( f + g = x^2 + |x| \)
- (ii) \( f - g = x^2 - |x| \)
- (iii) \( fg = x^2 |x| \)
- (iv) \( 2f = 2x^2 \)
- (v) \( f^2 = x^4 \)
- (vi) \( f + 3 = x^2 + 3 \)