we are given
[tex]f(x)=\frac{3x-1}{x+4}[/tex]
(a)
We know that when denominator of any function is 0
then function will become undefined
so, denominator=0 will make function no solution
so, we set denominator =0
and then we solve for x
[tex]x+4=0[/tex]
[tex]x=-4[/tex]............Answer
(b)
Vertical shift:
Suppose, we want to shift y=f(x) function vertically by 'c' units
we add 'c' top y-value
so, new function will become
[tex]y=f(x)+c[/tex]
now, we have
If g(x) is a vertical shift of 4 units of f(x)
so, we can write our function as
[tex]g(x)=f(x)+4[/tex]
we can plug f(x)
[tex]g(x)=\frac{3x-1}{x+4}+4[/tex]
Comparison:
So, graph of f(x) is moved upside by 4 units to get graph of g(x)
(c)
We can set g(x)=8
and then we can solve for x
[tex]g(x)=\frac{3x-1}{x+4}+4=8[/tex]
Multiply both sides by x+4
[tex]\frac{3x-1}{x+4}\left(x+4\right)+4\left(x+4\right)=8\left(x+4\right)[/tex]
[tex]3x-1+4\left(x+4\right)=8\left(x+4\right)[/tex]
[tex]7x+15=8x+32[/tex]
[tex]7x=8x+17[/tex]
[tex]-x=17[/tex]
[tex]x=-17[/tex]..............Answer
