Graph of a Function

Question

Given $$ f(x) = \frac{1}{x – 2} + 5 \qquad x\neq 2 $$ Explain how the graph of $f$ can be obtained from the graph of $\frac1x$ by using appropriate translations. Include the image set of the function $f$.

Answer 1

Let $g(x) = \frac1x$, then $$ f(x) = g(x-2)+5 $$ That is, you basically do a change of coordinates from $y=g(x)$ to $\hat y = g(\hat x)$ where $$ \begin{cases} \hat x & = x-2 \\ \hat y & = y-5 \end{cases} $$ $\hat x$ accounts for a horizontal translation of $+2$, i.e. of two units to the right, and $\hat y$ for a vertical translation of $+5$, i.e. of two units upward. In the end the graph of $f(x)$ is that of $\frac1x$ translated by vector $(2,5)$.

Finally graphically deduce the image of $f(x)$ from that of $\frac1x$ which is clearly $\mathbb R\setminus 0$.

Answer 2

Have a nice graph:

(Large Version)