Find whether the following function are one-one or many -one & into or onto if `f:DtoR` where `D` is its domain
`f(x)=(1+x^(6))/(x^(3))`

Explanation

To determine whether the function \( f: D \to \mathbb{R} \) defined by \( f(x) = \frac{1 + x^6}{x^3} \) is one-one or many-one, and whether it is into or onto, we will follow these steps:

### Step 1: Check if the function is one-one or many-one

To check if the function is one-one, we need to see if \( f(x_1) = f(x_2) \) implies \( x_1 = x_2 \).

1. Start with the equation:
\[ f(x_1) = f(x_2) \]
This gives us:
\[ \frac{1 + x_1^6}{x_1^3} = \frac{1 + x_2^6}{x_2^3} \]

2. Cross-multiply to eliminate the fractions:
\[ (1 + x_1^6) x_2^3 = (1 + x_2^6) x_1^3 \]

3. Rearranging gives:
\[ x_1^3 - x_2^3 = \frac{1}{x_2^3} - \frac{1}{x_1^3} \]

4. Factor the left side:
\[ x_1^3 - x_2^3 = (x_1 - x_2)(x_1^2 + x_1 x_2 + x_2^2) \]

5. The right side simplifies to:
\[ \frac{x_1^3 - x_2^3}{x_1^3 x_2^3} \]

6. Setting both sides equal gives:
\[ (x_1 - x_2)(x_1^2 + x_1 x_2 + x_2^2) = \frac{(x_1 - x_2)(x_1^2 + x_1 x_2 + x_2^2)}{x_1^3 x_2^3} \]

7. If \( x_1 \neq x_2 \), we can cancel \( (x_1 - x_2) \) from both sides, leading to:
\[ x_1^3 x_2^3 = 1 \]
This indicates that \( x_1 \) and \( x_2 \) can be different values that satisfy \( x_1 x_2 = 1 \).

Thus, the function is **many-one**.

### Step 2: Check if the function is into or onto

1. The codomain of \( f \) is \( \mathbb{R} \). We need to find if there are any values in \( \mathbb{R} \) that cannot be achieved by \( f(x) \).

2. We observe that:
\[ f(x) = \frac{1 + x^6}{x^3} \]
The numerator \( 1 + x^6 \) is always positive for all real \( x \), and since \( x^3 \) can be both positive and negative, \( f(x) \) can never be zero.

3. To find the range of \( f(x) \):
- As \( x \to 0 \), \( f(x) \to \infty \).
- As \( x \to \infty \), \( f(x) \to \infty \).
- As \( x \to -\infty \), \( f(x) \to -\infty \) (since \( x^3 \) becomes negative).

4. However, \( f(x) \) can never equal zero, as shown above.

Since there are values in the codomain (specifically, 0) that are not in the range of \( f(x) \), the function is **into**.

### Conclusion

- The function \( f(x) = \frac{1 + x^6}{x^3} \) is **many-one** and **into**.

---