Algebra Level 5

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

The range is simply R{ac}\mathbb{R}-\left\{\frac{a}{c}\right\}. In other words, the range is R{the coefficient of x in the numeratorthe coefficient of x in the denominator}\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)=3x+25x+9f(x)=\dfrac{3x+2}{5x+9} is R{35}\mathbb{R}- \left\{\frac{3}{5}\right\}.

Now consider the following statements.

[1][1]. The range of f(x)=9x54x+3f(x)=\dfrac{9x-5}{4x+3} is R{94}\mathbb{R}- \left\{\frac{9}{4}\right\}.

[2][2]. The range of f(x)=26x1691x56f(x)=\dfrac{26x-16}{91x-56} is R{2691}\mathbb{R}- \left\{\frac{26}{91}\right\}.

[3][3]. The range of f(x)=2x617x41f(x)=\dfrac{-2x-6}{17x-41} is R{217}\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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