Given the initial function,
[tex]f(x)=2^x[/tex]
In general, a vertical stretch/compression is expressed by
[tex]f(x)\to k\cdot f(x)[/tex]
If k>1, the function gets a vertical stretch; on the other hand, if 0Therefore, in our case,
[tex]g_1(x)=\frac{1}{3}f(x)\to\text{vertical compression by a factor of 1/3}[/tex]
A vertical shift is given by the following formula
[tex]\begin{gathered} f(x)+k \\ k>0\to\text{shifted up} \\ k<0\to\text{shifted down} \end{gathered}[/tex]
In our case,
[tex]g(x)=g_1(x)-7\to\text{vertical shift down by 7 units.}[/tex]
Therefore, the answers are B and D.
