Respuesta :
Using the asymptote concept, the function with a vertical asymptote at x = 3 and an horizontal asymptote at [tex]y = -\frac{1}{2}[/tex] is given by:
[tex]f(x) = -\frac{x}{2(x - 3)}[/tex]
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:
[tex]f(x) = \frac{g(x)}{x - 3}[/tex]
The horizontal asymptote at y = -1/2 means that:
[tex]\lim_{x \rightarrow \infty} f(x) = -\frac{1}{2}[/tex]
Which happens if [tex]g(x) = -\frac{x}{2}[/tex], hence the function is:
[tex]f(x) = -\frac{x}{2(x - 3)}[/tex]
More can be learned about asymptotes at https://brainly.com/question/16948935
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