Step 1

a)Start from the graph of the parent function \(\displaystyle{f{{\left({x}\right)}}}={{\log}_{{{3}}}{x}}\)

As we can see the given function \(\displaystyle{y}={{\log}_{{{3}}}{x}}+{2}\) can be expressed in terms of the parent function f as \(\displaystyle{y}={f{{\left({x}\right)}}}+{2}\)

This indicates that the graph of the function \(\displaystyle{y}={{\log}_{{{3}}}{x}}+{2}\) will be the same as the graph of the parent function \(\displaystyle{f{{\left({x}\right)}}}={{\log}_{{{3}}}{x}}\) shifted 2 units upward.

See the graphs in the picture below:

Step 2

b) The domain of the function \(\displaystyle{y}={{\log}_{{{3}}}{x}}+{2}\) is the interval: \(\displaystyle{\left({0},+\infty\right)}\)

The range of the function \(\displaystyle{y}={{\log}_{{{3}}}{x}}+{2}\) is the interval \(\displaystyle{\left(-\infty,+\infty\right)}\)

c) The vertical asymptote of the graph of this function is the line \(\displaystyle{x}={0}\)

a)Start from the graph of the parent function \(\displaystyle{f{{\left({x}\right)}}}={{\log}_{{{3}}}{x}}\)

As we can see the given function \(\displaystyle{y}={{\log}_{{{3}}}{x}}+{2}\) can be expressed in terms of the parent function f as \(\displaystyle{y}={f{{\left({x}\right)}}}+{2}\)

This indicates that the graph of the function \(\displaystyle{y}={{\log}_{{{3}}}{x}}+{2}\) will be the same as the graph of the parent function \(\displaystyle{f{{\left({x}\right)}}}={{\log}_{{{3}}}{x}}\) shifted 2 units upward.

See the graphs in the picture below:

Step 2

b) The domain of the function \(\displaystyle{y}={{\log}_{{{3}}}{x}}+{2}\) is the interval: \(\displaystyle{\left({0},+\infty\right)}\)

The range of the function \(\displaystyle{y}={{\log}_{{{3}}}{x}}+{2}\) is the interval \(\displaystyle{\left(-\infty,+\infty\right)}\)

c) The vertical asymptote of the graph of this function is the line \(\displaystyle{x}={0}\)