Let `f(x) = e^(x) - x` and `g (x) = x^(2) - x, AA x in R`. Then the set of all `x in R` where the function `h(x) = (f o g) (x)` in increasing is : `

Options

A. `[-1, (-1)/(2)] cap [(1)/(2), oo)`
B. `[0,(1)/(2)] cup [1, oo)`
C. `[0, oo)`
D. `[(-1)/(2), 0] cup [1,oo)`

Answer

To solve the problem, we need to determine the intervals where the function \( h(x) = (f \circ g)(x) \) is increasing. Here, \( f(x) = e^x - x \) and \( g(x) = x^2 - x \).

### Step-by-Step Solution:

1. **Define the composite function**:
\[ h(x) = f(g(x)) = f(x^2 - x) = e^{(x^2 - x)} - (x^2 - x) \]

2. **Differentiate \( h(x) \)**:
To find where \( h(x) \) is increasing, we need to compute the derivative \( h'(x) \):
\[ h'(x) = \frac{d}{dx}(e^{(x^2 - x)}) - \frac{d}{dx}(x^2 - x) \]
Using the chain rule for the first term:
\[ h'(x) = e^{(x^2 - x)} \cdot (2x - 1) - (2x - 1) \]
Simplifying gives:
\[ h'(x) = (2x - 1)(e^{(x^2 - x)} - 1) \]

3. **Set the derivative greater than or equal to zero**:
For \( h(x) \) to be increasing, we need:
\[ h'(x) \geq 0 \implies (2x - 1)(e^{(x^2 - x)} - 1) \geq 0 \]

4. **Analyze the factors**:
- The factor \( 2x - 1 \) changes sign at \( x = \frac{1}{2} \).
- The factor \( e^{(x^2 - x)} - 1 \) is non-negative when \( e^{(x^2 - x)} \geq 1 \), which occurs when \( x^2 - x \geq 0 \).

5. **Solve \( x^2 - x \geq 0 \)**:
Factoring gives:
\[ x(x - 1) \geq 0 \]
The critical points are \( x = 0 \) and \( x = 1 \). Analyzing the intervals:
- \( (-\infty, 0) \): Positive
- \( (0, 1) \): Negative
- \( (1, \infty) \): Positive

Thus, \( x^2 - x \geq 0 \) for \( x \in (-\infty, 0] \cup [1, \infty) \).

6. **Combine intervals**:
We now have two cases:
- **Case 1**: \( 2x - 1 \geq 0 \) (i.e., \( x \geq \frac{1}{2} \)) and \( e^{(x^2 - x)} - 1 \geq 0 \) (i.e., \( x \in [1, \infty) \)).
- **Case 2**: \( 2x - 1 \leq 0 \) (i.e., \( x < \frac{1}{2} \)) and \( e^{(x^2 - x)} - 1 \leq 0 \) (i.e., \( x \in [0, \frac{1}{2}) \)).

The intervals where \( h'(x) \geq 0 \) are:
- From Case 1: \( [1, \infty) \)
- From Case 2: \( [0, \frac{1}{2}) \)

7. **Final intervals**:
Therefore, the set of all \( x \in \mathbb{R} \) where \( h(x) \) is increasing is:
\[ [0, \frac{1}{2}) \cup [1, \infty) \]