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What is the graph of the function f(x) = the quantity of negative x squared minus 2 x minus 2, all over x minus 2?


Group of answer choices

graph with vertical asymptote of x equals negative 2, and oblique asymptote of y equals negative x

graph with vertical asymptote of x equals 2, and oblique asymptote of y equals x

graph with vertical asymptote of x equals 2, and oblique asymptote of y equals negative x minus 4

graph with vertical asymptote of x equals 5, and oblique asymptote of y equals negative x minus 4

Sagot :

Given:

The function is

[tex]f(x)=\dfrac{x^2-2x-2}{x-2}[/tex]

To find:

The vertical asymptote and oblique asymptote.

Solution:

We have,

[tex]f(x)=\dfrac{x^2-2x-2}{x-2}[/tex]

To find vertical asymptote, equate denominator equal to 0.

[tex]x-2=0[/tex]

[tex]x=2[/tex]

So, the vertical asymptote is [tex]x=2[/tex].

In the given function degree of numerator is greater than denominator so, their is an oblique asymptote. To find oblique asymptote divide the numerator by denominator.

Dividing [tex]x^2-2x-2[/tex] by [tex]x-2[/tex] using synthetic division, we get

2 |  1   -2    -2

          2     0

--------------------------

     1    0    -2

-------------------------

Here, starting elements of bottom row represent coefficient of quotient and last element of bottom row represents the remainder.

[tex]Quotient=x, Remainder=-2[/tex]

Since, quotient is x, therefore, the oblique asymptote is [tex]y=x[/tex].

Therefore, the correct option is B.

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