In the following items, you will analyze how several transformations affect the graph of the function f(x)=frac{1}{x}. Investigate the graphs of f(x)=

Transformations of functions
In the following items, you will analyze how several transformations affect the graph of the function $$\displaystyle{f{{\left({x}\right)}}}={\frac{{{1}}}{{{x}}}}$$. Investigate the graphs of $$\displaystyle{f{{\left({x}\right)}}}={\frac{{{1}}}{{{x}}}},{g{{\left({x}\right)}}}={f{{\left({x}\right)}}}={\frac{{{1}}}{{{x}+{2}}}},{h}{\left({x}\right)}={\frac{{{1}}}{{{x}-{2}}}},{p}{\left({x}\right)}={\frac{{{1}}}{{{x}-{4}}}}\ \text{and}\ {z}{\left({x}\right)}={\frac{{{1}}}{{{x}^{{{2}}}+{1}}}}$$. If you use a graphing calculator, select a viewing window $$\displaystyle\pm{23.5}$$ for x and $$\displaystyle\pm{15.5}$$ for y. At what values in the domain did vertical asymptotes occur for each of the functions? Explain why the vertical asymptotes occur at these values.

Step 1 We graph f(x), as well as g(x) (dotted line), h(x) (dashed line), and p(x) (solid line). Step 2 We graph z(x): Step 3 We find that the vertical asymptotes are $$\displaystyle{f{{\left({x}\right)}}}:{x}={0}$$
$$\displaystyle{g{{\left({x}\right)}}}:{x}=-{2}$$
$$\displaystyle{h}{\left({x}\right)}:{x}={2}$$
$$\displaystyle{p}{\left({x}\right)}:{x}={4}$$
$$\displaystyle{z}{\left({x}\right)}:$$ None Each vertical asymptote is found by finding the value(s) of x for which the denominator equals zero, and therefore the function is undefined. Note that there is no such real value of x in the case of z(x).