Answer:
Vertical asymptote: [tex]x=3[/tex]
Horizontal asymptote: [tex]f(x) =2[/tex]
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:
[tex]f (x) = -\dfrac{1 }{ x-3} +2[/tex]
One root, [tex]x = 3.5[/tex]
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).
Roots of f(x) means the value of x where f(x) = 0
[tex]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[/tex]
One root, [tex]x = 3.5[/tex]
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) [tex]\rightarrow \infty[/tex]
For all, other values of [tex]x[/tex] , [tex]f(x)[/tex] 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 [tex]\frac{1}{x-3}[/tex].
Hence, Range of f(x) is all real numbers except 2.
Vertical Asymptote is the value of x, where value of f(x) [tex]\rightarrow \infty[/tex].
[tex]-\dfrac{1 }{ x-3} +2 \rightarrow \infty[/tex]
It is possible only when
[tex]x-3=0\\\Rightarrow x=3[/tex]
[tex]\therefore[/tex] vertical asymptote: [tex]x=3[/tex]
Horizontal Asymptote is the value of f(x) , where value of x [tex]\rightarrow \infty[/tex].
[tex]x\rightarrow \infty \Rightarrow \dfrac{1 }{ x-3} \rightarrow 0\\\therefore f(x) =-0+2 \\\Rightarrow f(x) =2[/tex]
[tex]\therefore[/tex] Horizontal asymptote: [tex]f(x) =2[/tex]
Please refer to the graph of given function as shown in the attached image.
