Find the Asymptotes e^x-5
The question asks for the identification of asymptotes of the function f(x) = e^x - 5. An asymptote refers to a line that the graph of a function approaches but never actually reaches as the independent variable (in this case x) approaches infinity or negative infinity. There are various types of asymptotes - vertical, horizontal, and oblique (slant). In the context of this function, you would typically look for horizontal and possibly vertical asymptotes. Horizontal asymptotes are typically found for functions as x approaches infinity or negative infinity, while vertical asymptotes are associated with values of x that cause the function to approach infinity or negative infinity (often where the function is undefined).
$e^{x} - 5$
Answer
Solution:
Step 1:
Identify the horizontal asymptote for the given exponential function. Since the function is $e^x - 5$, the horizontal asymptote occurs where the function levels off as $x$ approaches infinity. The horizontal asymptote is thus at $y = -5$.
Horizontal Asymptote: $y = -5$
Step 2:
There are no vertical asymptotes for this function. Exponential functions like $e^x$ do not have vertical asymptotes, as they are defined for all real numbers. Additionally, subtracting 5 does not introduce any vertical asymptotes.
Knowledge Notes:
Exponential functions, such as $f(x) = e^x$, are continuous and defined for all real numbers $x$. They have the following characteristics:
The base of an exponential function is a positive real number not equal to 1.
Exponential functions grow without bound as $x$ approaches infinity, and they approach zero as $x$ approaches negative infinity.
The horizontal asymptote of an exponential function of the form $f(x) = e^x + k$ or $f(x) = e^x - k$ is the line $y = k$. This is because as $x$ becomes very large or very small, the $e^x$ term dominates the behavior of the function, and the constant $k$ determines where the function levels off.
Exponential functions do not have vertical asymptotes because they are defined for all $x$ values and do not approach infinity for any finite value of $x$.
In the case of the function $f(x) = e^x - 5$, the horizontal asymptote is found by looking at the behavior of the function as $x$ approaches infinity. The term $e^x$ grows without bound, but since we are subtracting 5, the function will approach 5 units below wherever $e^x$ would be on the y-axis. Therefore, the horizontal asymptote is $y = -5$.
Understanding asymptotes is crucial for graphing functions and analyzing their behavior at the extremes. Horizontal asymptotes indicate the value that the function approaches as $x$ goes to infinity or negative infinity. Vertical asymptotes, which do not exist for exponential functions, would indicate values of $x$ where the function approaches infinity.
