# Shortcut!

Algebra Level 5

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.

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

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

$$. 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.

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