Find the range of the following functions:
`f(x)=(e^(x))/(1+absx),xge0` [.] denotes the greatest integer function.

Answer

To find the range of the function \( f(x) = \frac{e^x}{1 + |x|} \) for \( x \geq 0 \), we will follow these steps:

### Step 1: Analyze the function
Since \( x \geq 0 \), we can simplify the absolute value:
\[ f(x) = \frac{e^x}{1 + x} \]

### Step 2: Determine the behavior as \( x \) approaches 0
Calculate \( f(0) \):
\[ f(0) = \frac{e^0}{1 + 0} = \frac{1}{1} = 1 \]

### Step 3: Determine the behavior as \( x \) approaches infinity
Next, we analyze the limit of \( f(x) \) as \( x \) approaches infinity:
\[ \lim_{x \to \infty} f(x) = \lim_{x \to \infty} \frac{e^x}{1 + x} \]
As \( x \) approaches infinity, \( e^x \) grows much faster than \( 1 + x \), so:
\[ \lim_{x \to \infty} f(x) = \infty \]

### Step 4: Check for critical points
To find the minimum and maximum values of \( f(x) \), we will take the derivative and set it to zero:
\[ f'(x) = \frac{(e^x)(1 + x) - e^x(1)}{(1 + x)^2} = \frac{e^x \cdot x}{(1 + x)^2} \]
Setting \( f'(x) = 0 \):
\[ e^x \cdot x = 0 \]
Since \( e^x > 0 \) for all \( x \), we have \( x = 0 \) as the only critical point.

### Step 5: Evaluate the function at the critical point
We already calculated:
\[ f(0) = 1 \]

### Step 6: Determine the nature of the critical point
To determine if this critical point is a minimum or maximum, we can check the sign of \( f'(x) \):
- For \( x > 0 \), \( f'(x) > 0 \) (since \( e^x > 0 \) and \( x > 0 \)), indicating that \( f(x) \) is increasing for \( x > 0 \).

### Step 7: Conclusion about the range
Since \( f(x) \) is increasing for \( x \geq 0 \) and starts from \( f(0) = 1 \) and approaches infinity as \( x \) approaches infinity, the range of \( f(x) \) is:
\[ [1, \infty) \]