`f(x)=1/abs(x), g(x)=sqrt(x^(-2))`

Answer

To determine if the functions \( f(x) = \frac{1}{|x|} \) and \( g(x) = \sqrt{x^{-2}} \) are identical, we need to check three necessary conditions: the domains, the ranges, and whether \( f(a) = g(a) \) for all \( a \) in the common domain.

### Step 1: Find the Domain of \( f(x) \)

The function \( f(x) = \frac{1}{|x|} \) has a denominator of \( |x| \). Since the absolute value of \( x \) can never be zero (as it would make the denominator undefined), the domain of \( f(x) \) is all real numbers except 0.

**Domain of \( f(x) \):**
\[ D_f = \{ x \in \mathbb{R} : x \neq 0 \} \]

### Step 2: Find the Range of \( f(x) \)

Since \( |x| \) is always positive for \( x \neq 0 \), \( f(x) \) will always yield positive values. Therefore, the range of \( f(x) \) is all positive real numbers.

**Range of \( f(x) \):**
\[ R_f = \{ y \in \mathbb{R} : y > 0 \} \]

### Step 3: Find the Domain of \( g(x) \)

The function \( g(x) = \sqrt{x^{-2}} \) can be rewritten as \( g(x) = \frac{1}{\sqrt{x^2}} \). The square root function is defined for all real numbers except where the argument is negative. Since \( x^2 \) is always non-negative and only equals zero when \( x = 0 \), the domain of \( g(x) \) is also all real numbers except 0.

**Domain of \( g(x) \):**
\[ D_g = \{ x \in \mathbb{R} : x \neq 0 \} \]

### Step 4: Find the Range of \( g(x) \)

Similar to \( f(x) \), since \( \sqrt{x^2} \) is always positive for \( x \neq 0 \), \( g(x) \) will also yield positive values. Thus, the range of \( g(x) \) is all positive real numbers.

**Range of \( g(x) \):**
\[ R_g = \{ y \in \mathbb{R} : y > 0 \} \]

### Step 5: Check if \( f(a) = g(a) \) for all \( a \) in the common domain

For any \( a \neq 0 \):
\[ f(a) = \frac{1}{|a|} \quad \text{and} \quad g(a) = \frac{1}{\sqrt{a^2}} = \frac{1}{|a|} \]
Thus, \( f(a) = g(a) \).

### Conclusion

Since the domains \( D_f \) and \( D_g \) are equal, the ranges \( R_f \) and \( R_g \) are equal, and \( f(a) = g(a) \) for all \( a \) in the common domain, we conclude that the functions \( f(x) \) and \( g(x) \) are identical.

**Final Result:**
\[ f(x) \text{ and } g(x) \text{ are identical functions.} \]

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