a) The given function is
![f(x)=\frac{x^2-4}{x^4+x^3-4x^2-4}](https://tex.z-dn.net/?f=f%28x%29%3D%5Cfrac%7Bx%5E2-4%7D%7Bx%5E4%2Bx%5E3-4x%5E2-4%7D)
The domain refers to all values of x for which the function is defined.
The function is defined for
![x^4+x^3-4x^2-4\ne0](https://tex.z-dn.net/?f=x%5E4%2Bx%5E3-4x%5E2-4%5Cne0)
This implies that;
![x\ne -2.69,x\ne 1.83](https://tex.z-dn.net/?f=x%5Cne%20-2.69%2Cx%5Cne%201.83)
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.
![x^4+x^3-4x^2-4=0](https://tex.z-dn.net/?f=x%5E4%2Bx%5E3-4x%5E2-4%3D0)
This implies that;
![x= -2.69,x=1.83](https://tex.z-dn.net/?f=x%3D%20-2.69%2Cx%3D1.83)
c) The roots are the x-intercepts of the graph.
To find the roots, we equate the function to zero and solve for x.
![\frac{x^2-4}{x^4+x^3-4x^2-4}=0](https://tex.z-dn.net/?f=%5Cfrac%7Bx%5E2-4%7D%7Bx%5E4%2Bx%5E3-4x%5E2-4%7D%3D0)
![\Rightarrow x^2-4=0](https://tex.z-dn.net/?f=%5CRightarrow%20x%5E2-4%3D0)
![x^2=4](https://tex.z-dn.net/?f=x%5E2%3D4)
![x=\pm \sqrt{4}](https://tex.z-dn.net/?f=x%3D%5Cpm%20%5Csqrt%7B4%7D)
![x=\pm2](https://tex.z-dn.net/?f=x%3D%5Cpm2)
The roots are ![x=-2,x=2](https://tex.z-dn.net/?f=x%3D-2%2Cx%3D2)
d) The y-intercept is where the graph touches the y-axis.
To find the y-inter, we substitute;
into the function
![f(0)=\frac{0^2-4}{0^4+0^3-4(0)^2-4}](https://tex.z-dn.net/?f=f%280%29%3D%5Cfrac%7B0%5E2-4%7D%7B0%5E4%2B0%5E3-4%280%29%5E2-4%7D)
![f(0)=\frac{-4}{-4}=1](https://tex.z-dn.net/?f=f%280%29%3D%5Cfrac%7B-4%7D%7B-4%7D%3D1)
e) to find the horizontal asypmtote, we take limit to infinity
![lim_{x\to \infty}\frac{x^2-4}{x^4+x^3-4x^2-4}=0](https://tex.z-dn.net/?f=lim_%7Bx%5Cto%20%5Cinfty%7D%5Cfrac%7Bx%5E2-4%7D%7Bx%5E4%2Bx%5E3-4x%5E2-4%7D%3D0)
The horizontal asymtote is ![y=0](https://tex.z-dn.net/?f=y%3D0)
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.