Let `f: R -{0} to R` be defined by `f(x) =x + 1/x`, then range of
`g(x) = (f(x))^(4) - f(x^(4)) - 4(f(x))^(2)`, is

Options

A. `R-(2)`
B. `{2}`
C. `{-2}`
D. `R - {2}`

Answer

To find the range of the function \( g(x) = (f(x))^4 - f(x^4) - 4(f(x))^2 \), where \( f(x) = x + \frac{1}{x} \), we will follow these steps:

### Step 1: Understand the function \( f(x) \)

The function \( f(x) = x + \frac{1}{x} \) is defined for all real numbers \( x \) except \( x = 0 \).

### Step 2: Determine the range of \( f(x) \)

To find the range of \( f(x) \), we can analyze its behavior.

1. **Finding the minimum value of \( f(x) \)**:
- We can use the AM-GM inequality:
\[ x + \frac{1}{x} \geq 2 \quad \text{for } x > 0 \]
and
\[ x + \frac{1}{x} \leq -2 \quad \text{for } x < 0 \]
- Thus, the minimum value of \( f(x) \) is \( 2 \) when \( x = 1 \) and \( -2 \) when \( x = -1 \).

2. **Conclusion about the range of \( f(x) \)**:
- Therefore, the range of \( f(x) \) is \( (-\infty, -2] \cup [2, \infty) \).

### Step 3: Find \( f(x^4) \)

Next, we need to compute \( f(x^4) \):
\[ f(x^4) = x^4 + \frac{1}{x^4} \]

### Step 4: Express \( g(x) \)

Now we can substitute \( f(x) \) and \( f(x^4) \) into \( g(x) \):
\[ g(x) = (f(x))^4 - f(x^4) - 4(f(x))^2 \]
Substituting \( f(x) = x + \frac{1}{x} \):
\[ g(x) = \left(x + \frac{1}{x}\right)^4 - \left(x^4 + \frac{1}{x^4}\right) - 4\left(x + \frac{1}{x}\right)^2 \]

### Step 5: Simplify \( g(x) \)

1. **Calculate \( (f(x))^2 \)**:
\[ (f(x))^2 = \left(x + \frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2} \]
2. **Calculate \( (f(x))^4 \)**:
\[ (f(x))^4 = \left(x^2 + 2 + \frac{1}{x^2}\right)^2 \]
3. **Calculate \( f(x^4) \)**:
\[ f(x^4) = x^4 + \frac{1}{x^4} \]

### Step 6: Substitute and simplify

After substituting and simplifying, we find that:
\[ g(x) = -2 \]

### Step 7: Determine the range of \( g(x) \)

Since \( g(x) \) simplifies to the constant value \( -2 \), the range of \( g(x) \) is simply:
\[ \text{Range of } g(x) = \{-2\} \]

### Final Answer:
The range of \( g(x) \) is \( \{-2\} \).

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