Question

For y=2+ \log_{3}x. a) Use transformations of the graphs of y=\log_{2}x

Transformations of functions
ANSWERED
asked 2021-08-14
For \(\displaystyle{y}={2}+{{\log}_{{{3}}}{x}}\).
a) Use transformations of the graphs of \(\displaystyle{y}={{\log}_{{{2}}}{x}}\ {\quad\text{and}\quad}\ {y}={{\log}_{{{3}}}{x}}\) o graph the given functions.
b) Write the domain and range in interval notation.
c) Write an equation of the asymptote.

Answers (1)

2021-08-15
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:
image 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 \(\displaystyle{x}={0}\)
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