Shortcut!

There is a really fast way of finding the ranges of functions of the form \(f(x)=\dfrac{ax+b}{cx+d}\) where the \(f(x)\) is defined from \(\mathbb{R}-\left\{\frac{-d}{c}\right\}\) to \(\mathbb{R}\).

The range is simply \(\mathbb{R}-\left\{\frac{a}{c}\right\}\). In other words, the range is \(\mathbb{R}-\left\{\dfrac{\text{the coefficient of } x\ \text{in the numerator}}{\text{the coefficient of } x \ \text{in the denominator}}\right\}\) [verify this!].

For example, the range of \(f(x)=\dfrac{3x+2}{5x+9}\) is \(\mathbb{R}- \left\{\frac{3}{5}\right\}\).

Now consider the following statements.

\([1]\). The range of \(f(x)=\dfrac{9x-5}{4x+3}\) is \(\mathbb{R}- \left\{\frac{9}{4}\right\}\).

\([2]\). The range of \(f(x)=\dfrac{26x-16}{91x-56}\) is \(\mathbb{R}- \left\{\frac{26}{91}\right\}\).

\([3]\). The range of \(f(x)=\dfrac{-2x-6}{17x-41}\) is \(\mathbb{R}- \left\{\frac{-2}{17}\right\}\).

Which of them are correct?

This problem is from the set "MCQ Is Not As Easy As 1-2-3". You can see the rest of the problems here.