The function f(x) = x^{2} + bx + c,where b an | vedclass

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(D) The given function is $f(x) = x^{2} + bx + c$. This is a quadratic function representing a parabola. For a function to be one-to-one,$f(x_1) = f(x

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neither one-to-one nor onto mapping

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(D) The given function is $f(x) = x^{2} + bx + c$. This is a quadratic function representing a parabola. For a function to be one-to-one,$f(x_1) = f(x_2)$ must imply $x_1 = x_2$. Here,$f(x) = (x + \frac{b}{2})^2 + (c - \frac{b^2}{4})$. Since the square term is always non-negative,$f(x_1) = f(x_2)$ does not necessarily imply $x_1 = x_2$ (e.g.,$f(x) = x^2$ gives $f(1) = f(-1) = 1$),so it is many-to-one. Furthermore,the range of this function is $[c - \frac{b^2}{4}, \infty)$,which is not equal to the codomain $\mathbb{R}$ (assuming the codomain is $\mathbb{R}$),so it is not onto. Therefore,the function is neither one-to-one nor onto.

The function $f(x) = x^{2} + bx + c$,where $b$ and $c$ are real constants,describes:

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