Question

Find the vertical and horizontal asymptotes, domain, range, and roots of f (x) = -1 / x-3 +2.

Answer 1

Vertical asymptote:
`x=3`

Horizontal asymptote:
`f(x) =2`

Domain of f(x) is all real numbers except 3.

Range of f(x) is all real numbers except 2.

Explanation:

Given:

Function:

`f (x) = -(1 )/( x-3) +2`

One root,
`x = 3.5`

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).

Roots of f(x) means the value of x where f(x) = 0

`0= -(1 )/( x-3) +2\\\Rightarrow 2= (1 )/( x-3)\\\Rightarrow 2x-2 * 3=1\\\Rightarrow 2x=7\\\Rightarrow x = 3.5`

One root,
`x = 3.5`

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`

For all, other values of
`x` ,
`f(x)` 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
`(1)/(x-3)`.

Hence, Range of f(x) is all real numbers except 2.

Vertical Asymptote is the value of x, where value of f(x)
`\rightarrow \infty`.

`-(1 )/( x-3) +2 \rightarrow \infty`

It is possible only when

`x-3=0\\\Rightarrow x=3`

`\therefore` vertical asymptote:
`x=3`

Horizontal Asymptote is the value of f(x) , where value of x
`\rightarrow \infty`.

`x\rightarrow \infty \Rightarrow (1 )/( x-3) \rightarrow 0\\\therefore f(x) =-0+2 \\\Rightarrow f(x) =2`

`\therefore` Horizontal asymptote:
`f(x) =2`

Please refer to the graph of given function as shown in the attached image.