Answer:
Vertical asymptote: ![x=3](https://tex.z-dn.net/?f=x%3D3)
Horizontal asymptote: ![f(x) =2](https://tex.z-dn.net/?f=f%28x%29%20%3D2)
Domain of f(x) is all real numbers except 3.
Range of f(x) is all real numbers except 2.
Step-by-step explanation:
Given:
Function:
![f (x) = -\dfrac{1 }{ x-3} +2](https://tex.z-dn.net/?f=f%20%28x%29%20%3D%20-%5Cdfrac%7B1%20%7D%7B%20x-3%7D%20%2B2)
One root, ![x = 3.5](https://tex.z-dn.net/?f=x%20%3D%203.5)
To find:
Vertical and horizontal asymptote, domain, range and roots of f(x).
Solution:
First of all, let us find the roots of f(x).
<em>Roots of f(x) means the value of x where f(x) = 0</em>
![0= -\dfrac{1 }{ x-3} +2\\\Rightarrow 2= \dfrac{1 }{ x-3}\\\Rightarrow 2x-2 \times 3=1\\\Rightarrow 2x=7\\\Rightarrow x = 3.5](https://tex.z-dn.net/?f=0%3D%20-%5Cdfrac%7B1%20%7D%7B%20x-3%7D%20%2B2%5C%5C%5CRightarrow%202%3D%20%5Cdfrac%7B1%20%7D%7B%20x-3%7D%5C%5C%5CRightarrow%202x-2%20%5Ctimes%203%3D1%5C%5C%5CRightarrow%202x%3D7%5C%5C%5CRightarrow%20x%20%3D%203.5)
One root, ![x = 3.5](https://tex.z-dn.net/?f=x%20%3D%203.5)
Domain of f(x) i.e. the values that we give as input to the function and there is a value of f(x) defined for it.
For x = 3, the value of f(x) ![\rightarrow \infty](https://tex.z-dn.net/?f=%5Crightarrow%20%5Cinfty)
For all, other values of
,
is defined.
Hence, Domain of f(x) is all real numbers except 3.
Range of f(x) i.e. the values that are possible output of the function.
f(x) = 2 is not possible in this case because something is subtracted from 2. That something is
.
Hence, Range of f(x) is all real numbers except 2.
Vertical Asymptote is the value of x, where value of f(x)
.
![-\dfrac{1 }{ x-3} +2 \rightarrow \infty](https://tex.z-dn.net/?f=-%5Cdfrac%7B1%20%7D%7B%20x-3%7D%20%2B2%20%5Crightarrow%20%5Cinfty)
It is possible only when
![x-3=0\\\Rightarrow x=3](https://tex.z-dn.net/?f=x-3%3D0%5C%5C%5CRightarrow%20x%3D3)
vertical asymptote: ![x=3](https://tex.z-dn.net/?f=x%3D3)
Horizontal Asymptote is the value of f(x) , where value of x
.
![x\rightarrow \infty \Rightarrow \dfrac{1 }{ x-3} \rightarrow 0\\\therefore f(x) =-0+2 \\\Rightarrow f(x) =2](https://tex.z-dn.net/?f=x%5Crightarrow%20%5Cinfty%20%5CRightarrow%20%5Cdfrac%7B1%20%7D%7B%20x-3%7D%20%5Crightarrow%200%5C%5C%5Ctherefore%20f%28x%29%20%3D-0%2B2%20%5C%5C%5CRightarrow%20f%28x%29%20%3D2)
Horizontal asymptote: ![f(x) =2](https://tex.z-dn.net/?f=f%28x%29%20%3D2)
Please refer to the graph of given function as shown in the attached image.