Find the Asymptotes e^x-6
The question is asking to determine the asymptotes of the function \( e^x - 6 \). An asymptote is a line that the graph of a function approaches as the value of x increases or decreases without bounds. There are different types of asymptotes such as vertical, horizontal, and oblique (slant). The goal here is to identify any such lines for the given function.
$e^{x} - 6$
Answer
Solution:
Step 1:
Identify the horizontal asymptote for the given exponential function. Since the function is $e^x - 6$, the horizontal asymptote is the constant term that the function approaches as $x$ goes to positive or negative infinity. Therefore, the horizontal asymptote is at $y = -6$.
Horizontal Asymptote:
The horizontal asymptote for the function $e^x - 6$ is given by the equation $y = -6$.
Step 2:
There are no vertical asymptotes for exponential functions since they do not have values of $x$ for which the function goes to infinity. Thus, the function $e^x - 6$ does not have any vertical asymptotes.
Knowledge Notes:
Exponential functions are of the form $f(x) = a \cdot b^x + c$, where $a$, $b$, and $c$ are constants, and $b$ is the base of the exponential function. The function $e^x - 6$ is a specific case where $a=1$, $b=e$ (Euler's number, approximately equal to 2.71828), and $c=-6$.
Horizontal Asymptotes: A horizontal asymptote of a function is a horizontal line $y = k$ that the graph of the function approaches as $x$ tends to positive or negative infinity. For exponential functions, the horizontal asymptote is determined by the constant term $c$. If $c$ is present, the horizontal asymptote is $y = c$. If $c$ is not present, the horizontal asymptote is $y = 0$.
Vertical Asymptotes: A vertical asymptote is a vertical line $x = k$ that the graph of the function approaches as $x$ approaches $k$. Exponential functions do not have vertical asymptotes because they are defined for all real numbers and do not approach infinity for any finite value of $x$.
Behavior of Exponential Functions: Exponential functions grow rapidly as $x$ increases when the base $b > 1$. When $x$ is very large, the term $e^x$ dominates, and the constant term $-6$ becomes insignificant, which is why the function approaches the line $y = -6$ but never actually reaches it.
Understanding the behavior of exponential functions and their asymptotes is important in various fields such as mathematics, physics, biology, and economics, where these functions are used to model growth and decay processes.
