Find the Domain and Range of f(x)=(ax-b)/(cx-d)

Find the Domain and Range of $f(x)=\frac{ax-b}{cx-d}$

Question: Find the domain and range of the real valued function: $$ f(x)=\frac{ax-b}{cx-d} $$

Solution

Given: $$ f(x)=\frac{ax-b}{cx-d} $$

The denominator cannot be zero.

Therefore, $$ cx-d\ne0 $$

$$ x\ne\frac{d}{c} $$

Hence, the domain is: $$ \mathbb{R}-\left\{\frac{d}{c}\right\} $$

Let $$ y=\frac{ax-b}{cx-d} $$

Cross multiply: $$ y(cx-d)=ax-b $$

$$ cxy-dy=ax-b $$

$$ x(cy-a)=dy-b $$

$$ x=\frac{dy-b}{cy-a} $$

For $x$ to exist, $$ cy-a\ne0 $$

Therefore, $$ y\ne\frac{a}{c} $$

Hence, the range is: $$ \mathbb{R}-\left\{\frac{a}{c}\right\} $$

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