If a function `f:RtoR` is defined by `f(x)=(x^(2)-5)/(x^(2)+4)`, then f is

Options

A. one-one but not onto
B. onto but not one-one
C. neither one-one nor onto
D. a bijection

Answer

To determine the properties of the function \( f: \mathbb{R} \to \mathbb{R} \) defined by

\[ f(x) = \frac{x^2 - 5}{x^2 + 4}, \]

we need to check if the function is one-to-one (1-1) and onto.

### Step 1: Check if the function is one-to-one (1-1)

A function is one-to-one if \( f(x_1) = f(x_2) \) implies \( x_1 = x_2 \).

1. Assume \( f(x_1) = f(x_2) \):
\[ \frac{x_1^2 - 5}{x_1^2 + 4} = \frac{x_2^2 - 5}{x_2^2 + 4}. \]

2. Cross-multiply:
\[ (x_1^2 - 5)(x_2^2 + 4) = (x_2^2 - 5)(x_1^2 + 4). \]

3. Expanding both sides:
\[ x_1^2 x_2^2 + 4x_1^2 - 5x_2^2 - 20 = x_2^2 x_1^2 + 4x_2^2 - 5x_1^2 - 20. \]

4. Simplifying gives:
\[ 4x_1^2 - 5x_2^2 = 4x_2^2 - 5x_1^2. \]

5. Rearranging terms:
\[ 9x_1^2 = 9x_2^2. \]

6. Dividing by 9:
\[ x_1^2 = x_2^2. \]

7. This implies:
\[ x_1 = x_2 \quad \text{or} \quad x_1 = -x_2. \]

Since \( x_1 = -x_2 \) provides a second solution, the function is **not one-to-one**.

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

A function is onto if the range of the function is equal to its codomain.

1. Set \( y = f(x) \):
\[ y = \frac{x^2 - 5}{x^2 + 4}. \]

2. Rearranging gives:
\[ y(x^2 + 4) = x^2 - 5. \]

3. This leads to:
\[ yx^2 + 4y - x^2 + 5 = 0. \]

4. Rearranging terms:
\[ (y - 1)x^2 + 4y + 5 = 0. \]

5. For this quadratic equation in \( x \) to have real solutions, the discriminant must be non-negative:
\[ D = b^2 - 4ac = 0^2 - 4(y - 1)(4y + 5) \geq 0. \]

6. Simplifying the discriminant:
\[ -4(y - 1)(4y + 5) \geq 0. \]

7. This implies:
\[ (y - 1)(4y + 5) \leq 0. \]

8. Finding the roots:
- \( y - 1 = 0 \) gives \( y = 1 \).
- \( 4y + 5 = 0 \) gives \( y = -\frac{5}{4} \).

9. Testing intervals:
- The function is negative between \( -\frac{5}{4} \) and \( 1 \).

Thus, the range of \( f \) is \( \left[-\frac{5}{4}, 1\right] \), which is not equal to the codomain \( \mathbb{R} \). Therefore, the function is **not onto**.

### Conclusion

The function \( f(x) = \frac{x^2 - 5}{x^2 + 4} \) is neither one-to-one nor onto.

### Final Answer

The correct option is: **neither 1-1 nor onto**.

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