Ans(a):
Given function is [tex]f(x)=\frac{3x-1}{x+4}[/tex]
we know that any rational function is not defined when denominator is 0 so that means denominator x+4 can't be 0
so let's solve
x+4≠0 for x
x≠0-4
x≠-4
Hence at x=4, function can't have solution.
Ans(b):
We know that vertical shift occurs when we add something on the right side of function so vertical shift by 4 units means add 4 to f(x)
so we get:
g(x)=f(x)+4
[tex]g(x)=\frac{3x-1}{x+4}+4[/tex]
We may simplify this equation but that is not compulsory.
Comparision:
Graph of g(x) will be just 4 unit upward than graph of f(x).
Ans(c):
To find value of x when g(x)=8, just plug g(x)=8 in previous equation
[tex]8=\frac{3x-1}{x+4}+4[/tex]
[tex]8-4=\frac{3x-1}{x+4}[/tex]
[tex]4=\frac{3x-1}{x+4}[/tex]
[tex]4(x+4)=(3x-1)[/tex]
[tex]4x+16=3x-1[/tex]
4x-3x=-1-16
x=-17
Hence final answer is x=-17
