Find whether the following functions are one-one or not.
`f:RtoR`, defined by `f(x)=x^(3),x inR`

Answer

To determine whether the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = x^3 \) is one-one (injective), we will follow these steps:

### Step 1: Understand the Definition of One-One Function
A function \( f \) is said to be one-one if for any two different inputs \( p \) and \( q \) in the domain, the outputs are different. Mathematically, this means if \( f(p) = f(q) \), then it must follow that \( p = q \).

### Step 2: Assume \( f(p) = f(q) \)
Let's assume that \( f(p) = f(q) \) for some \( p, q \in \mathbb{R} \). According to our function, we have:
\[ f(p) = p^3 \quad \text{and} \quad f(q) = q^3 \]
Thus, we can write:
\[ p^3 = q^3 \]

### Step 3: Solve the Equation
From the equation \( p^3 = q^3 \), we can take the cube root of both sides:
\[ p = q \]
This shows that if the outputs are equal, the inputs must also be equal.

### Step 4: Conclusion
Since we have shown that \( f(p) = f(q) \) implies \( p = q \), we conclude that the function \( f(x) = x^3 \) is indeed a one-one function.

### Final Answer
The function \( f(x) = x^3 \) is a one-one function.

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