Answer:
1. x=0
2. x>0
3. X-intercept: (0.5,0) Y-intercept: undefined
4. Compressed horizontally by a factor of 0.25 and shifted 1 unit down.
Step-by-step explanation:
1. The given parent function is ![f(x)=\log_2x](https://tex.z-dn.net/?f=f%28x%29%3D%5Clog_2x)
The transformed function is ![g(x)=\log_2(x*4)-1](https://tex.z-dn.net/?f=g%28x%29%3D%5Clog_2%28x%2A4%29-1)
The vertical asymptote is x=0.
We find this by equating the argument to zero. 4x=0----->x=0
2. The domain is all values of x that make the function defined.
The domain is ![x\:>\:0](https://tex.z-dn.net/?f=x%5C%3A%3E%5C%3A0)
3. To find x-intercept we substitute g(x)=y=0 and solve for x.
![\log_2(x*4)-1=0](https://tex.z-dn.net/?f=%5Clog_2%28x%2A4%29-1%3D0)
![\log_2(x*4)=1](https://tex.z-dn.net/?f=%5Clog_2%28x%2A4%29%3D1)
![(x*4)=2^1](https://tex.z-dn.net/?f=%28x%2A4%29%3D2%5E1)
![4x=2](https://tex.z-dn.net/?f=4x%3D2)
x=0.5
The x-intercept is (0.5,0)
The y-intercept is undefined because we have to substitute x=0, which is not in the domain of g(x).
4. We can rewrite the transformed function in terms of f(x) as
![g(x)=f(4x)-1](https://tex.z-dn.net/?f=g%28x%29%3Df%284x%29-1)
This means there is a horizontal compression by
and shift 1 unit down of the parent function.