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: \( \begin{array}{ll}\text { (d) one-one } & \text { (b) onto } \\ \text { (d) bijective } & \text { (d) neither one-one nor onto }\end{array} \)

Let's analyze the function \( f: \mathbb{R}_+ \rightarrow [-5, \infty) \) defined by \( f(x) = 9x^2 + 6x - 5 \), where \( \mathbb{R}_+ \) is the set of all non-negative real numbers. ### 1. **One-One (Injective)** A function is **one-one** if each element of the domain maps to a unique element in the codomain. - **Derivative Analysis:** \( f'(x) = 18x + 6 \) For \( x \geq 0 \), \( f'(x) \geq 6 \), which means the function is **strictly increasing** on \( \mathbb{R}_+ \). - **Conclusion:** Since the function is strictly increasing, it is **one-one**. ### 2. **Onto (Surjective)** A function is **onto** if every element in the codomain has a pre-image in the domain. - **Range of \( f(x) \):** \( f(0) = 9(0)^2 + 6(0) - 5 = -5 \) As \( x \) increases, \( f(x) \) increases without bound, approaching \( \infty \). - **Conclusion:** The range of \( f(x) \) is \( [-5, \infty) \), which exactly matches the codomain. Therefore, \( f \) is **onto**. ### 3. **Bijective** A function is **bijective** if it is both one-one and onto. - **Conclusion:** Since \( f \) is both one-one and onto, it is **bijective**. **Final Answer:** \( f \) is **bijective**. **Answer:** \( f \) is **bijective**.