Respuesta :

Answer:

Vertical Asymptote:

[tex]x=1[/tex]

Horizontal asymptote:

it does not exist

Step-by-step explanation:

we are given

[tex]f(x)=log_3(x-1)[/tex]

Vertical asymptote:

we know that vertical asymptotes are values of x where f(x) becomes +inf or -inf

we know that any log becomes -inf when value inside log is zero

so, we can set value inside log to zero

and then we can solve for x

[tex]x-1=0[/tex]

we get

[tex]x=1[/tex]

Horizontal asymptote:

we know that

horizontal asymptote is a value of y when x is +inf or -inf

For finding horizontal asymptote , we find lim x-->inf or -inf

[tex]\lim_{x \to \infty}  f(x)= \lim_{x \to \infty}log_3(x-1)[/tex]

[tex]\lim_{x \to \infty}  f(x)=log_3(\infty-1)[/tex]

[tex]\lim_{x \to \infty}  f(x)=undefined[/tex]

so, it does not exist

tifanihayyu

The asymptote for the graph of this logarithmic function is at x = 1

Further explanation

An asymptote of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity. Whereas the logarithm is the inverse function to exponentiation and any exponential function can be expressed in logarithmic form.

First we solve [tex]log_3(x-1) = 0[/tex]

Raise both sides of the equation by the base of the logarithm:

[tex]3^(log_3(x-1)) = 3^0[/tex]

Because any non-zero number raised to 0 equals 1, the right side simplifies to 1

By using the property of logarithms that [tex]b^{(logbx)} = x,[/tex] where the left side simplifies to x-1

The equation is simply: x - 1 = 1

So x = 2.

The domain of a logarithmic function is the set of all positive real numbers. For example f(x) = log x also has an asymptote at x =0. But, since our function is log (x-1), we will move the asymptote to the right by 1 unit. Thus, x = 1.  Which explains that the graph will never touch at x=1 which will be the vertical asymptote.

Learn more

  1. Learn more about asymptote  https://brainly.com/question/10730051
  2. Learn more about logarithmic function https://brainly.com/question/1447265
  3. Learn more about the graph of this logarithmic function https://brainly.com/question/9132850

Answer details

Grade:  9

Subject:  mathematics

Chapter:  logarithmic function

Keywords:  logarithmic function, asymptote, graph, curve,  the inverse function

Otras preguntas