Let f be a function that is differentiable everywhere and that has the follwong properties :
(i) `f(x) gt 0`
(ii) `f'(0) = -1`
(iii) `f(-x) = (1)/(f(x))and f(x+h)=f(x).f(h)`
A standard result is `(f'(x))/(f(x))dx = log|f(x)| + C`
Range of f(x) is

Options

A. R
B. `R - {0}`
C. `R^(+)`
D. (0, e)

Answer

To solve the problem step by step, we will analyze the properties of the function \( f \) given in the question and derive its range.

### Step 1: Analyze the properties of \( f \)

We are given the following properties of the function \( f \):
1. \( f(x) > 0 \) for all \( x \).
2. \( f'(0) = -1 \).
3. \( f(-x) = \frac{1}{f(x)} \).
4. \( f(x + h) = f(x) \cdot f(h) \).

### Step 2: Determine \( f(0) \)

From property 3, substituting \( x = 0 \):
\[ f(-0) = f(0) = \frac{1}{f(0)} \]
This implies:
\[ f(0)^2 = 1 \quad \Rightarrow \quad f(0) = 1 \text{ (since \( f(x) > 0 \))} \]

### Step 3: Differentiate \( f(x) \)

Using property 4, we can express the derivative \( f'(x) \):
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} = \lim_{h \to 0} \frac{f(x)f(h) - f(x)}{h} \]
Factoring out \( f(x) \):
\[ = f(x) \lim_{h \to 0} \frac{f(h) - 1}{h} \]
The limit \( \lim_{h \to 0} \frac{f(h) - 1}{h} \) is simply \( f'(0) \) since \( f(0) = 1 \). Thus:
\[ f'(x) = f(x) f'(0) = f(x)(-1) = -f(x) \]

### Step 4: Analyze the sign of \( f'(x) \)

Since \( f(x) > 0 \) for all \( x \), it follows that:
\[ f'(x) < 0 \text{ for all } x \]
This means that \( f(x) \) is a decreasing function.

### Step 5: Determine the behavior of \( f(x) \)

Given that \( f(0) = 1 \) and \( f(x) \) is decreasing:
- As \( x \to -\infty \), \( f(x) \) must approach a value greater than \( 0 \) (since \( f(x) > 0 \)).
- As \( x \to +\infty \), \( f(x) \) must approach a value less than \( 1 \).

### Step 6: Conclude the range of \( f(x) \)

Since \( f(x) \) is continuous and decreasing from \( 1 \) to some positive value as \( x \) moves from \( 0 \) to \( +\infty \) and approaches a value greater than \( 0 \) as \( x \) moves to \( -\infty \), we can conclude that the range of \( f(x) \) is:
\[ (0, 1] \]

### Final Answer

The range of \( f(x) \) is \( (0, 1] \).

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