If $f:R\rightarrow R$ be defined as $f(x)=x^{4}$ then

Question

Accepted Answer

To determine why Option 4 is the correct answer, we need to analyze the function $ f(x) = x^4 $ and its properties regarding being one-one (injective) and onto (surjective).

1. **One-One (Injective):**  
   A function is one-one if different inputs produce different outputs. In mathematical terms, $ f(x_1) = f(x_2) $ implies $ x_1 = x_2 $. For the function $ f(x) = x^4 $, consider two different values, $ x_1 = 1 $ and $ x_2 = -1 $. We have:
   $$
   f(1) = 1^4 = 1 \quad \text{and} \quad f(-1) = (-1)^4 = 1
   $$
   Here, $ f(1) = f(-1) $ but $ 1 \neq -1 $. This shows that the function is not one-one because different inputs can produce the same output.

2. **Onto (Surjective):**  
   A function is onto if every possible output in the codomain has a corresponding input in the domain. For $ f(x) = x^4 $, the codomain is all real numbers $ \mathbb{R} $. However, the range of $ f(x) = x^4 $ is only non-negative real numbers $[0, \infty)$ because any real number raised to the fourth power is non-negative. For example, there is no real $ x $ such that $ f(x) = -1 $. Thus, the function is not onto because not every real number is covered by the function's output.

Given these analyses, $ f(x) = x^4 $ is neither one-one nor onto. Therefore, Option 4 is correct.

**Explanation of why other options are incorrect:**

- **Option 1 (f is one-one onto):**  
  As shown, $ f(x) = x^4 $ is not one-one because different inputs can yield the same output. It is also not onto because not all real numbers are achievable as outputs. Hence, this option is incorrect.

- **Option 2 (f is many-one onto):**  
  While $ f(x) = x^4 $ is indeed many-one (as demonstrated by $ f(1) = f(-1) $), it is not onto because it does not cover all real numbers in the codomain. Therefore, this option is incorrect.

- **Option 3 (f is one-one but not onto):**  
  As established, the function is not one-one, so this option is incorrect.

In conclusion, the correct characterization of the function $ f(x) = x^4 $ is that it is neither one-one nor onto, making Option 4 the correct answer.