# 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.

$$[1]$$. 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.

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