Let the function \(f:\mathbb{R}\to\mathbb{R}\) be defined by \(f(x)=|3x+2|-6\). Which of the following statements is correct?

Correct answer: फलन न तो एक-एक (one-one) है और न ही onto; इसका परिसर \([-6,\infty)\) है

The function is not one-one because different inputs can produce the same output. For example, \(f(0)=|2|-6=-4\) and \(f\left(-\frac{4}{3}\right)=|-2|-6=-4\). Also, since \(|3x+2|\ge 0\), we have \(f(x)\ge -6\); hence values in \((-\infty,-6)\) are never obtained, so the function is not onto \(\mathbb{R}\). Therefore, option C is correct. Option A is wrong because the function is not one-one, while option B is wrong because it is not onto. Exam tip: For an absolute-value function, test one-one behaviour using symmetric inputs and test onto behaviour by finding its range.