Question

If the function \(f:\mathbb{R}\to\mathbb{R}\) is defined as \(f(x)=x^2-2x-3\), then \(f\) is:

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

Hint:

For quadratic functions \(ax^2+bx+c\), they are never one-one over \(\mathbb{R}\) and are onto \(\mathbb{R}\) only if the range covers all real numbers.

Answer 1

The Correct Option is C

Step 1: Check if \(f(x)\) is one-one.
A function is one-one if for any \(x_1 \neq x_2\), \(f(x_1) \neq f(x_2)\).
Here, \(f(x)=x^2-2x-3\) is a quadratic function, which is a parabola opening upwards.
A parabola is not one-one over \(\mathbb{R}\) because two different \(x\)-values can give the same \(f(x)\) value.
Hence, \(f(x)\) is

not one-one.

Step 2: Check if \(f(x)\) is onto.
A function \(f:\mathbb{R}\to\mathbb{R}\) is onto if for every \(y\in\mathbb{R}\), there exists \(x\in\mathbb{R}\) such that \(f(x)=y\).
The minimum value of \(f(x)\) occurs at \(x=-\frac{b}{2a} = \frac{2}{2} = 1\).
\(f(1)=1^2-2\cdot 1-3 = 1-2-3=-4\)
Thus, the range of \(f\) is \([-4,\infty)\).
Since \(\mathbb{R}\) includes values less than -4, \(f\) is

not onto.

Step 3: Conclusion.
Since \(f\) is neither one-one nor onto, the correct classification is:
\[ \boxed{\text{neither one-one nor onto}} \]