2. Let \( f: R, \rightarrow[-5, x) \) be defined as \( f(x)=9 x^{2}+6 x-5 \), where \( R_{+} \), is the set of all non-negative \( r \) n numbers. Then, \( f \) is: \( \begin{array}{ll}\text { (d) one-one } & \text { (b) onto } \\ \text { (d) bijective } & \text { (d) neither one-one nor onto }\end{array} \)

To determine the nature of the function \( f(x) = 9x^2 + 6x - 5 \) defined on \( R_{+} \), we first examine its behavior. Since it is a quadratic function with a positive leading coefficient, it opens upwards and has a minimum point, indicating that it is not one-one over its entire range. Additionally, we find that it’s not onto since its range is restricted to \( [-5, x) \), so it cannot cover all real numbers. Also, exploring its vertex, the minimum value occurs at \( x = -\frac{b}{2a} = -\frac{6}{2 \cdot 9} = -\frac{1}{3} \), which is outside the non-negative domain. Therefore, the range begins at its minimum value, confirming it does not meet the criteria for being onto either. Thus, \( f \) is neither one-one nor onto.