Consider the function `f: R to R` defined as `f(x) = x^(3)`.
Assertion (A): f(x) is a one-one function.
Reason (R): f(x) is a one-one function if co-domain = range.

Options

A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is NOT the correct explanation of A
C. A is true but R is false
D. A is false and R is True

Answer

To solve the question, we need to analyze the function \( f(x) = x^3 \) and determine whether it is a one-one function and evaluate the assertion and reason provided.

### Step 1: Understanding the function
The function given is \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = x^3 \).

### Step 2: Checking if \( f(x) \) is one-one
A function is said to be one-one (injective) if for every pair of distinct inputs, the outputs are also distinct. In mathematical terms, if \( f(a) = f(b) \) implies \( a = b \), then the function is one-one.

Let's assume:
\[ f(a) = f(b) \]
This means:
\[ a^3 = b^3 \]

Taking the cube root of both sides, we get:
\[ a = b \]

Since the only way for \( f(a) \) to equal \( f(b) \) is if \( a \) equals \( b \), we conclude that \( f(x) = x^3 \) is indeed a one-one function.

### Step 3: Analyzing the assertion (A)
The assertion states that \( f(x) \) is a one-one function. From our analysis in Step 2, we have confirmed that this assertion is true.

### Step 4: Analyzing the reason (R)
The reason states that \( f(x) \) is a one-one function if the co-domain equals the range. However, this statement is misleading. A function can be one-one without the co-domain being equal to the range. The function \( f(x) = x^3 \) is one-one regardless of whether the co-domain (which is all real numbers) equals the range (which is also all real numbers). Therefore, the reason is false.

### Conclusion
- Assertion (A) is true: \( f(x) = x^3 \) is a one-one function.
- Reason (R) is false: A function can be one-one without the co-domain being equal to the range.

Thus, the final answer is that the assertion is true, and the reason is false.