(1) Vertical asymptote: [tex]x=-4[/tex]
(2) Domain: [tex]x>-4[/tex]
(3) X intercept: [tex](-2,0)[/tex] and Y intercept : [tex](0,1)[/tex]
(4) The function g(x) is shifted 4 units to the left and shifted 1 unit down.
Explanation:
The parent function is [tex]f(x)=\log _{2} x[/tex]
The transformed function is [tex]g(x)=\log _{2}(x+4)-1[/tex]
(1) Vertical asymptote:
The vertical asymptote of a function can be determined by equating
[tex]x+4=0[/tex]
Thus, [tex]x=-4[/tex]
The vertical asymptote is [tex]x=-4[/tex]
(2) Domain:
The domain of a function is the set of all independent x-values.
[tex]x+4>0[/tex]
Thus, [tex]x>-4[/tex]
The domain of a function is [tex]x>-4[/tex]
(3) X and Y intercepts:
To determine the x intercept, let us substitute y=0 in [tex]g(x)=\log _{2}(x+4)-1[/tex]
[tex]\begin{equation}\begin{aligned}\log _{2}(x+4)-1 &=0 \\\log _{2}(x+4) &=1 \\x+4 &=2^{1} \\x &=-2\end{aligned}[/tex]
Thus, the x intercept is [tex](-2,0)[/tex]
To determine the y intercept, let us substitute x=0 in [tex]g(x)=\log _{2}(x+4)-1[/tex]
[tex]\begin{equation}\begin{aligned}y &=\log _{2}(0+4)-1 \\&=\log _{2} 4-1 \\&=2-1 \\&=1\end{aligned}[/tex]
Thus, the y intercept is [tex](0,1)[/tex]
(4) To determine the transformation:
The transformed function [tex]g(x)=\log _{2}(x+4)-1[/tex] is shifted 4 units to the left and shifted 1 unit downwards.
