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 [tex]f(x)=\log_2x[/tex]
The transformed function is [tex]g(x)=\log_2(x*4)-1[/tex]
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 [tex]x\:>\:0[/tex]
3. To find x-intercept we substitute g(x)=y=0 and solve for x.
[tex]\log_2(x*4)-1=0[/tex]
[tex]\log_2(x*4)=1[/tex]
[tex](x*4)=2^1[/tex]
[tex]4x=2[/tex]
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
[tex]g(x)=f(4x)-1[/tex]
This means there is a horizontal compression by [tex]\frac{1}{4}[/tex] and shift 1 unit down of the parent function.
The three features of the functions are:
The equation of the function is given as:
[tex]g(x)=\log_2(x*4)-1[/tex]
Set the argument to 0
[tex]x*4 = 0[/tex]
Divide both sides by 4
[tex]x= 0[/tex]
Hence, the vertical asymptote is [tex]x= 0[/tex]
In (a), we have the vertical asymptote to be [tex]x= 0[/tex]
This means that, the domain of the function is:
[tex]x > 0[/tex]
In (a), we have the vertical asymptote to be [tex]x= 0[/tex]
This means that, the function is undefined at x = 0
So, the y-intercept is undefined
Set y = 0 to calculate the x-intercept
[tex]g(x)=\log_2(x*4)-1[/tex]
[tex]\log_2(x*4)-1 = 0[/tex]
Add 1 to both sides
[tex]\log_2(x*4)=1[/tex]
Apply law of logarithm
[tex]x*4=2^1[/tex]
[tex]x*4=2[/tex]
Divide both sides by 4
[tex]x=\frac 12[/tex]
Hence, the x-intercept is 1/2
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