Exercise 42 Page 513

Let's start by reviewing the parent function, $y=\frac{1}{x}.$ This function has two branches and its asymptotes lie on the $x\text{-}$ and $y\text{-axes}.$

Its domain is all real numbers except for $x=0$ where the function is undefined. Similarly, the range is all real values except for $0,$ since there is no $x\text{-}$value for which $\frac{1}{x}=0.$ Is for these reason that the graph has asymptotic behavior. We can translate our function vertically by $\text{adding}$ the constant $k$ to the parent function. If $k$ is positive, the function gets translated upwards by $k$ units. But if $k<0$ then it gets translated downwards by $|k|$ units. Note that, just as the function, the horizontal asymptote translates either down or up, accordingly. We can find an example below. With this translation, the asymptotic behavior of the function around the horizontal asymptote gets shifted, and now the range is all real numbers except $3$ — since there is no $x\text{-}$value for which $y_2=3.$ Similarly, we can translate the function horizontally by $\text{replacing}$ the $x$ variable with $x-h$ in the parent function. If $h$ is positive, the function gets translated $h$ units to the right. But, if $h<0$ it gets translated $|h|$ units to the left. Just as in the previous example, if the function is translated horizontally, to the right of to the left, its vertical asymptote is translated in the same way. We can see an example below. Because the vertical asymptote is shifted $3$ units to the right, the new domain is all real numbers except $3,$ where the function is now undefined. We can also perform both translations at once. In this case, both the domain and range are modified in the same way we already discussed above. In this exercise we are asked to describe how the domain and range change when $y=\frac{1}{x}$ gets translated $3$ units up and $5$ units to the left. To do this, we can just substitute $k=3$ and $h=\text{-}5.$ Following the ideas above and using the values mentioned we can find the domain and range for the translated function.