Determine the equation of the horizontal asymptote for the graph of this function, and state the domain and range of
this function.
-$$f ( x ) = \log ( x + 5 ) + 3$$

Question

Quizplus · Accepted Answer

The function has a vertical asymptote at $x=-5$, which means that as $x$ approaches $-5$ from either direction, the function approaches either positive or negative infinity. However, as $x$ becomes very large (either positive or negative), the value of $\log(x+5)$ becomes dominated by the $x$ term, which grows exponentially. Therefore, the overall function approaches $\infty$ as $x$ approaches $\pm \infty$, and there is no horizontal asymptote. The domain of the function is $( -5 , \infty )$, since the argument of the logarithm must be positive, and the range is $( - \infty , \infty )$, since the range of the logarithmic function is all real numbers.

Determine the Equation of the Horizontal Asymptote for the Graph f(x)=log⁡(x+5)+3f ( x ) = \log ( x + 5 ) + 3f(x)=log(x+5)+3

Determine the Equation of the Horizontal Asymptote for the Graph $f ( x ) = \log ( x + 5 ) + 3$