Explanation:
We were given the function:
[tex]g(x)=-1+4^{x-1}[/tex]
We are to determine its domain, range and horizontal asymptote. This is shown below:
Domain:
[tex]\begin{gathered} g(x)=-1+4^{x-1} \\ 4^{x-1} \\ when:x=-10 \\ 4^{-10-1}=4^{-11} \\ when:x=1 \\ 4^^{1-1}=4^0=1 \\ when:x=20 \\ 4^{20-1}=4^{19} \\ \text{This shows us that the function is valid for every real number. This is written as:} \\ \left\{x|x∈R\right\} \end{gathered}[/tex]
Range:
[tex]\begin{gathered} g(x)=-1+4^{x-1} \\ \begin{equation*} -1+4^{x-1} \end{equation*} \\ when:x=-10 \\ =-1+4^{-10-1}\Rightarrow-1+4^{-11} \\ =-0.9999\approx-1 \\ when:x=1 \\ =-1+4^{1-1}\Rightarrow-1+4^0\Rightarrow-1+1 \\ =0 \\ when:x=5 \\ =-1+4^{5-1}\Rightarrow-1+4^4\Rightarrow-1+256 \\ =255 \\ \text{This shows us that the lowest value of ''y'' is -1. This is written as:} \\ \left\{y|y>−1\right\} \end{gathered}[/tex]
Horizontal asmyptote:
For exponential functions, the equation of the horizontal asymptote is given as:
[tex]y=-1[/tex]
