Which of the following pairs are equal functions ?

Options

A. `ln x^(2), 2ln x , x gt 0`
B. `(x^(2))/(x), x`
C. `|x|^(2), x^(2)`
D. `|x|^(3), x^(3)`

Answer

To determine which pairs of functions are equal functions, we need to check if their domains and ranges are the same. Let's analyze each pair step by step.

### Step 1: Understand Equal Functions
Equal functions are defined as functions that have the same domain and the same range. If \( f(x) \) and \( g(x) \) are two functions, they are equal if:
- Domain of \( f \) = Domain of \( g \)
- Range of \( f \) = Range of \( g \)

### Step 2: Analyze Each Pair of Functions

#### Pair 1: \( f(x) = \sqrt{x} \) and \( g(x) = x^2 \)
- **Domain of \( f(x) \)**: \( x \geq 0 \) (since square root is defined for non-negative numbers)
- **Domain of \( g(x) \)**: All real numbers \( x \) (since \( x^2 \) is defined for all real numbers)

Since the domains are not equal, \( f(x) \) and \( g(x) \) are **not equal functions**.

#### Pair 2: \( f(x) = \frac{1}{x} \) and \( g(x) = x \)
- **Domain of \( f(x) \)**: All real numbers except \( x = 0 \) (undefined at zero)
- **Domain of \( g(x) \)**: All real numbers \( x \)

Since the domains are not equal, \( f(x) \) and \( g(x) \) are **not equal functions**.

#### Pair 3: \( f(x) = |x| \) and \( g(x) = x^2 \)
- **Domain of \( f(x) \)**: All real numbers \( x \)
- **Domain of \( g(x) \)**: All real numbers \( x \)

Both functions have the same domain. Now, let's check the ranges:
- **Range of \( f(x) \)**: \( [0, \infty) \) (absolute value is always non-negative)
- **Range of \( g(x) \)**: \( [0, \infty) \) (since \( x^2 \) is also non-negative)

Since both the domain and range are the same, \( f(x) \) and \( g(x) \) are **equal functions**.

#### Pair 4: \( f(x) = |x| \) and \( g(x) = x^3 \)
- **Domain of \( f(x) \)**: All real numbers \( x \)
- **Domain of \( g(x) \)**: All real numbers \( x \)

Both functions have the same domain. Now, let's check the ranges:
- **Range of \( f(x) \)**: \( [0, \infty) \)
- **Range of \( g(x) \)**: All real numbers \( (-\infty, \infty) \) (since \( x^3 \) can take any real value)

Since the ranges are not the same, \( f(x) \) and \( g(x) \) are **not equal functions**.

### Conclusion
The only pair of functions that are equal is Pair 3: \( f(x) = |x| \) and \( g(x) = x^2 \).

### Final Answer
The correct option is Pair 3.

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