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

B) Vertical asymptote: $x = - 5$;
Domain: $( - 5 , \infty )$, Range: $( - \infty , \infty )$

C) Vertical asymptote: $x = 0$;
Domain: $( - \infty , 0 )$, Range: $( 0 , \infty )$

D) Vertical asymptote: $x = + 3$;
Domain: $( + 3 , \infty )$, Range: $( - \infty , \infty )$

Question

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

B) Vertical asymptote: $x = - 5$;
Domain: $( - 5 , \infty )$, Range: $( - \infty , \infty )$

C) Vertical asymptote: $x = 0$;
Domain: $( - \infty , 0 )$, Range: $( 0 , \infty )$

D) Vertical asymptote: $x = + 3$;
Domain: $( + 3 , \infty )$, Range: $( - \infty , \infty )$

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 $f ( x ) = \log ( x + 5 ) + 3$