Let f: R→R be the function defined by f (x) = x² ∀x ∈ R. Then show that f is neither one-one nor onto function.

To show that the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = x^2 \) for all \( x \in \mathbb{R} \) is neither one-one (injective) nor onto (surjective), we'll consider the definitions of these properties and analyze the function.

### 1. **Proving that \( f(x) = x^2 \) is not one-one:**

A function \( f \) is one-one (injective) if for every pair of distinct elements \( x_1, x_2 \in \mathbb{R} \), \( f(x_1) \neq f(x_2) \). In other words, if \( f(x_1) = f(x_2) \) implies \( x_1 = x_2 \), then \( f \) is injective.

Consider \( x_1 = 2 \) and \( x_2 = -2 \):

\[
f(2) = 2^2 = 4
\]
\[
f(-2) = (-2)^2 = 4
\]

Here, \( f(2) = f(-2) = 4 \), but \( 2 \neq -2 \). Therefore, \( f(x) = x^2 \) is not injective because two different inputs (2 and -2) yield the same output.

### 2. **Proving that \( f(x) = x^2 \) is not onto:**

A function \( f \) is onto (surjective) if for every element \( y \in \mathbb{R} \), there exists an element \( x \in \mathbb{R} \) such that \( f(x) = y \). In other words, every element in the codomain (which is \( \mathbb{R} \) in this case) must be covered by the function.

Consider \( y = -1 \in \mathbb{R} \). 

We need to find an \( x \in \mathbb{R} \) such that:

\[
f(x) = x^2 = -1
\]

However, there is no real number \( x \) such that \( x^2 = -1 \). The square of any real number is always non-negative (greater than or equal to 0), so it's impossible for \( f(x) = x^2 \) to produce a negative output.

Thus, the function does not cover all elements of \( \mathbb{R} \), meaning \( f(x) = x^2 \) is not surjective.

### **Conclusion:**
- \( f(x) = x^2 \) is not injective because different inputs can yield the same output (e.g., \( f(2) = f(-2) = 4 \)).
- \( f(x) = x^2 \) is not surjective because there are elements in the codomain \( \mathbb{R} \) that are not mapped by any element in the domain (e.g., \( -1 \) is not the square of any real number).

Thus, \( f(x) = x^2 \) is neither one-one nor onto.

Would you like further details or have any questions?

### Related Questions:
1. What are the conditions for a function to be both injective and surjective?
2. How can you modify the function \( f(x) = x^2 \) to make it injective?
3. What is the graphical interpretation of injectivity and surjectivity?
4. How do you prove that a function is bijective?
5. What are examples of functions that are both injective and surjective?

**Tip**: Graphing the function can often provide a visual understanding of why a function is not injective or surjective. For example, the parabola \( y = x^2 \) is symmetric, which explains the lack of injectivity, and it only covers non-negative \( y \)-values, which explains the lack of surjectivity.

Explore why the function f(x) = x^2 is neither one-one (injective) nor onto (surjective) through detailed analysis. Learn about the definitions of injective and surjective functions, and how they apply to real numbers. Suitable for undergraduate students studying functions and real analysis.

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