Question

Show all work to identify the asymptotes and state the end behavior of the function f(x)=3x/x-9

Answer 1

Using the asymptote concept, we have that:

• The vertical asymptote is x = 9.

• The horizontal asymptote is y = 3.

• The end behavior is that as $x \rightarrow \infty, y \rightarrow 3$.

What are the asymptotes of a function f(x)?

• The vertical asymptotes are the values of x which are outside the domain, which in a fraction are the zeroes of the denominator.

• The horizontal asymptote is the value of f(x) as x goes to infinity, as long as this value is different of infinity.

In this problem, the function is:

$f(x) = \frac{3x}{x - 9}$

For the vertical asymptote, we have that:

x - 9 = 0 -> x = 9.

For the horizontal asymptote:

$y = \lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} \frac{3x}{x - 9} = \lim_{x \rightarrow \infty} \frac{3x}{x} = \lim_{x \rightarrow \infty} 3 = 3$

Hence, the end behavior is that as $x \rightarrow \infty, y \rightarrow 3$.

More can be learned about asymptotes at brainly.com/question/16948935

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