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Question

# Using what you have learned about transformations of the graph $y=\frac{1}{x}$, can you find the equations of the asymptotes of the function $f(x)=\frac{p}{c x+d}+q$, and hence state the domain and range?

Solution

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Since the asymptotes of the graph of the function $y=\dfrac{1}{x}$ are $x=0$ and $y=0$, and in order to obtain the graph of the function $y=\dfrac{p}{cx+d}+q$, we use the horizontal translation by $\dfrac{c}{d}$ units therefore the vertical asymptote will also translate horizontally and it will be $x=-\dfrac{d}{c}$. The graph also vertically translate $q$ units which leads the horizontal asymptote to translate $q$ units vertical and the equation will be $y=q$.

• Vertical asymptote is $x=-\dfrac{d}{c}$ implies the domain is $\mathbb{R}-\left\{-\dfrac{d}{c}\right\}$.
• Horizontal asymptote is $y=q$ implies the range is $\mathbb{R}-\{q\}$.

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