Begin by graphing f(x)=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 functions domain and range. g(x)= -2log_{2}x

Begin by graphing f(x)=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 functions domain and range. g(x)= -2log_{2}x

Question
Transformations of functions
asked 2021-01-28
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 functions domain and range. \(\displaystyle{g{{\left({x}\right)}}}=\ -{2}{{\log}_{{{2}}}{x}}\)

Answers (1)

2021-01-29

Step 1 For the following Logarithmic function defined by 1) \(\displaystyle{f{{\left({x}\right)}}}=\ {{\log}_{{{2}}}{x}}\) Graphing Logarithmic function given in equation (1) requires setting up table of coordinates, so that

\(\begin{array}{|c|c|} \hline x & f(x)=y=\ \log_{2}x & (x, y) \\ \hline 1 & f(1)=y=\ \log_{2}(1)\Rightarrow\ 2^{y}=1=2^{0}\Rightarrow\ y=0 & (1, 0) \\ \hline 2 & f(2)=y=\ \log_{2}(2)\Rightarrow\ 2^{y}=2=2^{1}\Rightarrow\ y=1 & (2, 1) \\ \hline 4 & f(4)=y=\ \log_{2}(4)\Rightarrow\ 2^{y}=4=2^{2}\Rightarrow\ y=2 & (4, 2) \\ \hline 8 & f(8)=y=\ \log_{2}(8)\Rightarrow\ 2^{y}=8=2^{3}\Rightarrow\ y=3 & (8, 3) \\ \hline 16 & f(16)=y=\ \log_{2}(16)\Rightarrow\ 2^{y}=16=2^{4}\Rightarrow\ y=4 & (16, 4) \\ \hline 32 & f(32)=y=\ \log_{2}(32)\Rightarrow\ 2^{y}=32=2^{5}\Rightarrow\ y=5 & (32, 5) \\ \hline 64 & f(64)=y=\ \log_{2}(64)\Rightarrow\ 2^{y}=64=2^{6}\Rightarrow\ y=6 & (64, 6) \\ \hline \end{array}\)

Step 2 We plot the following points between (x, y) determined from the table of coordinates and connect them with the continuous curve which represent the Logarithmic function \(\displaystyle{f{{\left({x}\right)}}}={{\log}_{{{2}}}{x}}\) as shown in Figure (1). Figure (1) represent the graph of Logarithmic function \(f (x)=\ \log_{2}x\) image

Step 3 Note that: The y-axis which represented by the equation \(\displaystyle{x}={0}\) is the vertical asymptote, so that the curve approches, but never touches the positive portion of the y-axis as shown in figure (1). The domain of \(f(x)=\ \log_{2}(x)\ \text{is all positive real numbers}\ x\ \in\ (0,\ \infty)\ \text{andthe range is all real numbers}\ y\ \in\ (-\infty,\ \infty)\).

Step 4 To graph the Logarithmic function \(\displaystyle{g{{\left({x}\right)}}}=\ -{2}\ {{\log}_{{{2}}}{\left({x}\right)}},\ \text{first we reflect the graph of the function}\ {f{{\left({x}\right)}}}={{\log}_{{{2}}}{\left({x}\right)}}\ \text{about the x-axis because of the negative sign in}\ {g{{\left({x}\right)}}}=\ -{2}\ {{\log}_{{{2}}}{\left({x}\right)}}\ \text{then the Logarithmic function}\ {f{{\left({x}\right)}}}={{\log}_{{{2}}}{\left({x}\right)}},\ \text{transform to a new graph of-}\ {{\log}_{{{2}}}{\left({x}\right)}}\ \text{second we vertically stretch the graph of the function-}\ \ {{\log}_{{{2}}}{\left({x}\right)}},\ \text{then the Logarithmic function}\ {f{{\left({x}\right)}}}={{\log}_{{{2}}}{\left({x}\right)}}\ \text{transform to a new graph}\ {g{{\left({x}\right)}}}=\ -{2}{{\log}_{{{2}}}{\left({x}\right)}}\ \text{so that table of coordinates becomes}\).
\(\begin{array}{|c|c|} \hline x & g(x)=y=\ \frac{1}{2}\log_{2}x & (x, y) \\ \hline 1 & g(1)=y=\ -2 \log_{2}(1)\Rightarrow\ \frac{-y}{2}=\log_{2}(1)\ \Rightarrow\ 2\frac{-y}{2}=1=2^{0}\ \Rightarrow\ \frac{-y}{2}=0\ \Rightarrow\ y=0 & (1,\ 0) \\ \hline 2 & g(2)=y=\ -2\log_{2}(2)\ \Rightarrow\ \frac{-y}{2}=\log_{2}(2)\ \Rightarrow\ 2\frac{-y}{2}=2=2^{1}\ \Rightarrow\ \frac{-y}{2}=1\ \Rightarrow\ y=\ -2 & (2,\ -2)\\ \hline 4 & g(4)=y=\ -2\log_{2}(4)\ \Rightarrow\ \frac{-y}{2}=\log_{2}(4)\ \Rightarrow\ 2\frac{-y}{2}=4=2^{2}\ \Rightarrow\ \frac{-y}{2}=2\ \Rightarrow\ y=\ -4 & (4,\ -4) \\ \hline 8 & g(8)=y=\ -2\log_{2}(8)\ \Rightarrow\ \frac{-y}{2}=\log_{2}(8)\ \Rightarrow\ 2\frac{-y}{2}=8=2^{3}\ \Rightarrow\ \frac{-y}{2}=3\ \Rightarrow\ y=\ -6 & (8,\ -6) \\ \hline 16 & g(16)=y=\ -2\log_{2}(16)\ \Rightarrow\ \frac{-y}{2}=\log_{2}(16)\ \Rightarrow\ 2\frac{-y}{2}=16=2^{4}\ \Rightarrow\ \frac{-y}{2}=4\ \Rightarrow\ y=\ -8 & (16,\ -8) \\ \hline 32 & g(32)=y=\ -2\log_{2}(32)\ \Rightarrow\ \frac{-y}{2}=\log_{2}(32)\ \Rightarrow\ 2\frac{-y}{2}=32=2^{5}\ \Rightarrow\ \frac{-y}{2}=5\ \Rightarrow\ y=\ -10 & (32,\ -10) \\ \hline 64 & g(64)=y=\ -2\log_{2}(64)\ \Rightarrow\ \frac{-y}{2}=\log_{2}(64)\ \Rightarrow\ 2\frac{-y}{2}=64=2^{6}\ \Rightarrow\ \frac{-y}{2}=6\ \Rightarrow\ y=\ -12 & (64,\ -12) \\ \hline \end{array}\)

Step 5 First, we reflect the graph of the function \(\displaystyle{f{{\left({x}\right)}}}=\ {{\log}_{{{2}}}{\left({x}\right)}},\ \text{shown in figure (1), about the x-axis, then it transform to anew graph of}\ -{{\log}_{{{2}}}{\left({x}\right)}},\ \text{then the Logarithmic function}\ {f{{\left({x}\right)}}}=\ {{\log}_{{{2}}}{\left({x}\right)}}\ \text{transform to a new graph}\ {g{{\left({x}\right)}}}=\ -{2}{{\log}_{{{2}}}{\left({x}\right)}}\ \text{as shown in Figure (2)}.\) Figure (2) represent the graph of the Logarithmic function \(\displaystyle{g{{\left({x}\right)}}}=\ -{2}{{\log}_{{{2}}}{\left({x}\right)}}\) image

Step 6 Also note that: The y-axis which represented by the equation \(\displaystyle{x}={0}\) is the vertical asymptote, so that the curve approches, but never touches the positive portion of the y-axis as shown in figure (2). The domain of \(\displaystyle{g{{\left({x}\right)}}}=\ -{2}{{\log}_{{{2}}}{\left({x}\right)}}\ \text{is all positive real numbers in the interval}\ {x}\ \in\ {\left({0},\ \infty\right)}\ \text{and the range is all real numbers}\ {y}\ \in\ {\left(-\infty,\ \infty\right)}.\) Step 7 Combining Figure (1) and Figure (2) of the functions f(x) and g(x) in the same rectangular coordinate system, we obtain the new graph represented in the following figure. Figure (3) represent the graph of the functions \(\displaystyle{f{{\left({x}\right)}}}=\ {{\log}_{{{2}}}{\left({x}\right)}}\ \text{and}\ {g{{\left({x}\right)}}}=\ -{2}{{\log}_{{{2}}}{\left({x}\right)}}\) together. image

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