a) The given function is
[tex]f(x)=\frac{x^2-4}{x^4+x^3-4x^2-4}[/tex]
The domain refers to all values of x for which the function is defined.
The function is defined for
[tex]x^4+x^3-4x^2-4\ne0[/tex]
This implies that;
[tex]x\ne -2.69,x\ne 1.83[/tex]
b) The vertical asymptotes are x-values that makes the function undefined.
To find the vertical asymptote, equate the denominator to zero and solve for x.
[tex]x^4+x^3-4x^2-4=0[/tex]
This implies that;
[tex]x= -2.69,x=1.83[/tex]
c) The roots are the x-intercepts of the graph.
To find the roots, we equate the function to zero and solve for x.
[tex]\frac{x^2-4}{x^4+x^3-4x^2-4}=0[/tex]
[tex]\Rightarrow x^2-4=0[/tex]
[tex]x^2=4[/tex]
[tex]x=\pm \sqrt{4}[/tex]
[tex]x=\pm2[/tex]
The roots are [tex]x=-2,x=2[/tex]
d) The y-intercept is where the graph touches the y-axis.
To find the y-inter, we substitute;
[tex]x=0[/tex] into the function
[tex]f(0)=\frac{0^2-4}{0^4+0^3-4(0)^2-4}[/tex]
[tex]f(0)=\frac{-4}{-4}=1[/tex]
e) to find the horizontal asypmtote, we take limit to infinity
[tex]lim_{x\to \infty}\frac{x^2-4}{x^4+x^3-4x^2-4}=0[/tex]
The horizontal asymtote is [tex]y=0[/tex]
f) The greatest common divisor of both the numerator and the denominator is 1.
There is no common factor of the numerator and the denominator which is at least a linear factor.
Therefore the function has no holes.
g) The given function is a proper rational function.
There is no oblique asymptote.
See attachment for graph.
