Answer:
The x-intercept is 2.
The function has no y-intercept.
The vertical asymptote of the function is x=0.
The horizontal asymptote of the function is x=0.
The function has hole at x=4.
Step-by-step explanation:
The given function is
![f\left(x\right)=\dfrac{(3x-6)(x-4)}{x(x-4)}](https://tex.z-dn.net/?f=f%5Cleft%28x%5Cright%29%3D%5Cdfrac%7B%283x-6%29%28x-4%29%7D%7Bx%28x-4%29%7D)
Cancel out common factors.
![f\left(x\right)=\dfrac{3x-6}{x}](https://tex.z-dn.net/?f=f%5Cleft%28x%5Cright%29%3D%5Cdfrac%7B3x-6%7D%7Bx%7D)
(i) x-intercept.
Substitute f(x)=0, in the given function.
![0=\dfrac{3x-6}{x}](https://tex.z-dn.net/?f=0%3D%5Cdfrac%7B3x-6%7D%7Bx%7D)
![0=3x-6](https://tex.z-dn.net/?f=0%3D3x-6)
![-3x=-6](https://tex.z-dn.net/?f=-3x%3D-6)
![x=2](https://tex.z-dn.net/?f=x%3D2)
The x-intercept is 2.
(i) y-intercept.
Substitute x=0, in the given function.
![f\left(x\right)=\dfrac{3(0)-6}{(0)}=\infty](https://tex.z-dn.net/?f=f%5Cleft%28x%5Cright%29%3D%5Cdfrac%7B3%280%29-6%7D%7B%280%29%7D%3D%5Cinfty)
The function has no y-intercept.
(iii) Vertical asymptote.
Equate the denominator equal to 0.
![x=0](https://tex.z-dn.net/?f=x%3D0)
Therefore, the vertical asymptote of the function is x=0.
(iv) Horizontal asymptote.
If degree of numerator and denominator are same, then horizontal asymptote is
![y=\frac{\text{Leading coefficient of numerator}}{\text{Leading coefficient of denominator}}](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B%5Ctext%7BLeading%20coefficient%20of%20numerator%7D%7D%7B%5Ctext%7BLeading%20coefficient%20of%20denominator%7D%7D)
![y=\frac{3}{1}](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B3%7D%7B1%7D)
![y=3](https://tex.z-dn.net/?f=y%3D3)
Therefore, the horizontal asymptote of the function is x=0.
(v) End behavior
![f(x)\rightarrow 3\text{ as }x\rightarrow -\infty](https://tex.z-dn.net/?f=f%28x%29%5Crightarrow%203%5Ctext%7B%20as%20%7Dx%5Crightarrow%20-%5Cinfty)
![f(x)\rightarrow \infty\text{ as }\rightarrow 0^{-}](https://tex.z-dn.net/?f=f%28x%29%5Crightarrow%20%5Cinfty%5Ctext%7B%20as%20%7D%5Crightarrow%200%5E%7B-%7D)
![f(x)\rightarrow -\infty\text{ as }\rightarrow 0^{+}](https://tex.z-dn.net/?f=f%28x%29%5Crightarrow%20-%5Cinfty%5Ctext%7B%20as%20%7D%5Crightarrow%200%5E%7B%2B%7D)
![f(x)\rightarrow 3\text{ as }\rightarrow \infty](https://tex.z-dn.net/?f=f%28x%29%5Crightarrow%203%5Ctext%7B%20as%20%7D%5Crightarrow%20%5Cinfty%20)
(vi) holes
Equate the cancel factors equal to 0, to find the holes.
![x-4=0](https://tex.z-dn.net/?f=x-4%3D0)
![x=4](https://tex.z-dn.net/?f=x%3D4)
The function has hole at x=4.