Question

Which function has the following characteristics?

Answer 1

Using the asymptote concept, the function with a vertical asymptote at x = 3 and an horizontal asymptote at $y = -\frac{1}{2}$ is given by:

$f(x) = -\frac{x}{2(x - 3)}$

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.

The vertical asymptote at x = 3 means that x = 3 is a root of the denominator, hence:

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

The horizontal asymptote at y = -1/2 means that:

$\lim_{x \rightarrow \infty} f(x) = -\frac{1}{2}$

Which happens if $g(x) = -\frac{x}{2}$, hence the function is:

$f(x) = -\frac{x}{2(x - 3)}$

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

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