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An asymptote is a vertical horizontal or oblique line to which the graph of a function progressively approaches without ever touching it.

To answer this question we observe the graph. All the values of x and y must be identified for which the graph of the function tends to infinity.

It is observed that these values are:

x = -1

x = 3

y = 0

The first two corresponds to the equations of a vertical line. The third corresponds to horizontal line, the axis of x. It can be seen that although the graph of the function is very close to these values, it never "touches" them

The asymptotes to the considered function are given as:

  • Option 1: [tex]x= -1[/tex]  (vertical asymptote)
  • Option 3: [tex]x= 3[/tex] (vertical asymptote)
  • Option 5: [tex]y =0[/tex]  (horizontal asymptote)

When do we get vertical asymptote for a function?

Suppose that we have the function f(x) such that it is continuous for all input values < a or > a and have got the values of f(x) going  to infinity or -ve infinity (from either side of [tex]x = a[/tex]) as x goes near a , and being not defined at [tex]x = a[/tex], then at that point, there can be constructed a vertical line [tex]x = a[/tex] and it will be called as vertical asymptote for f(x) at [tex]x = a[/tex]

When do we get horizontal asymptote for a function?

The line [tex]y = a[/tex] is horizontal asymptote if the function f(x) tends to 'a' from upside of that line y = a, or from downside of that line.

For the given case, the line y = 0 is one of its horizontal asymptote.

If we make two straight lines at x = -1, and x = 3, we get two lines to which the graph of the function is going arbitrarily close, and going to +ve or -ve infinity.

Thus, they are its vertical asymptotes.

Therefore, the asymptotes to the considered function are given as:

  • Option 1: [tex]x= -1[/tex]  (vertical asymptote)
  • Option 3: [tex]x= 3[/tex] (vertical asymptote)
  • Option 5: [tex]y =0[/tex]  (horizontal asymptote)

Learn more about asymptotes here:

https://brainly.com/question/7327714

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