# Graph f and g in the same rectangular coordinate system. Use transformations of the graph of f to obtain the graph of g. Graph and give equations of all asymptotes. Use the graphs to determine each function's domain and range. f(x)=3^{x}=3^{x} - 1

Question
Transformations of functions
Graph f and g in the same rectangular coordinate system. Use transformations of the graph of f to obtain the graph of g. Graph and give equations of all asymptotes. Use the graphs to determine each function's domain and range. $$\displaystyle{f{{\left({x}\right)}}}={3}^{{{x}}}={3}^{{{x}}}\ -\ {1}$$

2021-02-11
$$\displaystyle{f{{\left({x}\right)}}}={3}^{{{x}}}\ {\quad\text{and}\quad}\ {g{{\left({x}\right)}}}={3}^{{{x}}}\ -\ {1}$$ Graph of g(x) can be obtained by translating the graph of f(x) by 1 unit downwards along the y-axis. $$\displaystyle\text{Domain of}\ {f{{\left({x}\right)}}}\ {i}{s}\ {x}\ \in\ {\left(-\infty,\ \infty\right)}$$
$$\displaystyle\text{Range of}{f{{\left({x}\right)}}}\ {i}{s}\ {\left({0},\ \infty\right)}$$
$$\displaystyle\text{Asymptote for the graph is}\ {y}={0}$$
$$\displaystyle\text{Domain of}\ {g{{\left({x}\right)}}}\ {i}{s}\ {x}\ \in\ {\left(-\infty,\ \infty\right)}$$
$$\displaystyle\text{Range of}{g{{\left({x}\right)}}}\ {i}{s}\ {\left(-{1},\ \infty\right)}$$
$$\displaystyle{P}{S}{K}\text{Asymptote for the graph is}\ {y}=\ -{1}$$ g(x) is the PURPLE curve, and f(x) is the RED curve

### Relevant Questions

Graph f and g in the same rectangular coordinate system. Use transformations of the graph f of to obtain the graph of g. Graph and give equations of all asymptotes. Use the graphs to determine each function's domain and range. $$\displaystyle{f{{\left({x}\right)}}}={2}^{{{x}}}\text{and}{g{{\left({x}\right)}}}={2}^{{{x}-{1}}}$$
Begin by graphing
$$\displaystyle{f{{\left({x}\right)}}}={2}^{{{x}}}.$$
Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs.
$$\displaystyle{g{{\left({x}\right)}}}={\frac{{{1}}}{{{2}}}}\ \cdot\ {2}^{{{x}}}$$
Begin by graphing
$$\displaystyle{f{{\left({x}\right)}}}={2}^{{{x}}}.$$
Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs.
$$\displaystyle{g{{\left({x}\right)}}}={2}^{{{x}\ +\ {2}}}$$
Begin by graphing
$$\displaystyle{f{{\left({x}\right)}}}={2}^{{{x}}}.$$
Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs.
$$\displaystyle{g{{\left({x}\right)}}}={2}^{{{x}}}={2}^{{{x}}}\ +\ {2}$$
Begin by graphing
$$\displaystyle{f{{\left({x}\right)}}}={2}^{{{x}}}$$
Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs.
$$\displaystyle{g{{\left({x}\right)}}}=-{2}^{{{x}}}$$
Begin by graphing
$$\displaystyle{f{{\left({x}\right)}}}={2}^{{{2}}}$$
Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs.
$$\displaystyle{g{{\left({x}\right)}}}={2}^{{-{x}}}$$
Begin by graphing $$\displaystyle{f{{\left({x}\right)}}}=\ {{\log}_{{{2}}}{x}}$$ Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each function's domain and range. $$\displaystyle{g{{\left({x}\right)}}}={\frac{{{1}}}{{{2}}}}\ {{\log}_{{{2}}}{x}}$$
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.
Begin by graphing $$\displaystyle{f{{\left({x}\right)}}}=\ {{\log}_{{{2}}}{x}}$$ Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each function's domain and range. $$\displaystyle{g{{\left({x}\right)}}}=\ -{2}\ {{\log}_{{{2}}}{x}}$$
$$\displaystyle{f{{\left({x}\right)}}}={{\log}_{{{2}}}{x}}$$
$$\displaystyle{g{{\left({x}\right)}}}={\frac{{{1}}}{{{2}}}}{{\log}_{{{2}}}{x}}$$