Question

The function $f:R\to R$ given by$f\left(x\right)=x^2+bx+c$, where $b$ and $c$ are real constants is

• A. a one-one function
• B. an onto function
• C. an onto but not a one-one function
• D. neither a one-one nor an onto function

Answer 1

The correct option is D neither a one-one nor an onto function

Given : $f\left(x\right)=x^2+bx+c$,
Since this is a quadratic function and we know, quadratic function is not a one-one in the given interval, because of its parabolic nature.
Also we know that the range of the above quadratic function is $[-\frac{(-b^2-4ac)}{4a},\infty )$ which is clearly a subset of the codomain $R$. Hence $f\left(x\right)$ is not an onto function.
$\therefore f\left(x\right)$ is neither a one-one nor an onto function.