if f(x) is a real valued function defined as `f(x)={min{|x|,1/x^2,1/x^3},x != 01, x=0` then range of `f(x)` is

Options

A. `(-oo, 1]`
B. `(-oo, 1] - {0}`
C. `[1, oo)`
D. R

Answer

To find the range of the function \( f(x) = \min\{|x|, \frac{1}{x^2}, \frac{1}{x^3}\} \) for \( x \neq 0 \) and \( f(0) = 1 \), we will analyze each component of the function separately and then determine the minimum value across all three components.

### Step 1: Analyze the function components

1. **For \( |x| \)**:
- This function is non-negative and increases as \( |x| \) increases.
- It approaches \( 0 \) as \( x \) approaches \( 0 \) from either side.

2. **For \( \frac{1}{x^2} \)**:
- This function is also non-negative and approaches \( \infty \) as \( x \) approaches \( 0 \).
- It decreases as \( |x| \) increases.

3. **For \( \frac{1}{x^3} \)**:
- This function behaves differently for positive and negative \( x \):
- For \( x > 0 \), \( \frac{1}{x^3} \) is positive and approaches \( \infty \) as \( x \) approaches \( 0 \).
- For \( x < 0 \), \( \frac{1}{x^3} \) is negative and approaches \( -\infty \) as \( x \) approaches \( 0 \).

### Step 2: Determine the minimum of the three functions

- For \( x > 0 \):
- \( |x| = x \)
- \( \frac{1}{x^2} \) is positive and decreases.
- \( \frac{1}{x^3} \) is positive and decreases.
- The minimum will be determined by comparing \( x \), \( \frac{1}{x^2} \), and \( \frac{1}{x^3} \).

- For \( x < 0 \):
- \( |x| = -x \)
- \( \frac{1}{x^2} \) is positive and decreases.
- \( \frac{1}{x^3} \) is negative and approaches \( -\infty \).
- The minimum will be determined by comparing \( -x \), \( \frac{1}{x^2} \), and \( \frac{1}{x^3} \).

### Step 3: Find the overall minimum

- As \( x \) approaches \( 0 \) from the positive side, \( \frac{1}{x^2} \) and \( \frac{1}{x^3} \) dominate, and the minimum approaches \( 0 \).
- As \( x \) approaches \( 0 \) from the negative side, \( \frac{1}{x^3} \) dominates and approaches \( -\infty \).

### Step 4: Determine the range of \( f(x) \)

- The function \( f(x) \) will take all values from \( -\infty \) to just below \( 0 \) as \( x \) approaches \( 0 \) from the left.
- At \( x = 0 \), \( f(0) = 1 \).
- Therefore, the range of \( f(x) \) is \( (-\infty, 0) \cup \{1\} \).

### Final Answer

The range of \( f(x) \) is \( (-\infty, 0) \cup \{1\} \).

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