Find the Asymptotes f(x)=4(4/5)^x
The given problem involves the exploration of the behavior of a given exponential function as the independent variable x approaches infinity or negative infinity. The goal is to determine the asymptotes of the function f(x) = 4*(4/5)^x. An asymptote of a function is a line that the graph of the function approaches but never touches as x becomes very large in positive or negative direction. The asymptotes can be vertical, horizontal, or oblique (slant). This particular question is asking you to identify and describe these asymptotic lines for the given exponential function.
$f \left(\right. x \left.\right) = 4 \left(\left(\right. \frac{4}{5} \left.\right)\right)^{x}$
Answer
Solution:
Step 1:
Identify the horizontal asymptote for the given exponential function. For $f(x) = 4\left(\frac{4}{5}\right)^x$, the horizontal asymptote is the value that the function approaches as $x$ goes to positive or negative infinity. The horizontal asymptote for this function is $y = 0$.
Step 2:
There are no vertical or oblique asymptotes for this function. Exponential functions like $f(x) = 4\left(\frac{4}{5}\right)^x$ do not have vertical or oblique asymptotes because they do not approach infinity or negative infinity for any finite value of $x$.
Knowledge Notes:
Exponential functions are of the form $f(x) = ab^x$, where $a$ is a constant, $b$ is the base of the exponential (with $b > 0$ and $b \neq 1$), and $x$ is the exponent. These functions have the following characteristics:
Horizontal Asymptote: The horizontal asymptote of an exponential function is determined by the constant term $a$. If $b > 1$, the function grows without bound as $x$ goes to positive infinity, and approaches $a$ as $x$ goes to negative infinity. If $0 < b < 1$, the function approaches $a$ as $x$ goes to positive infinity, and decreases without bound as $x$ goes to negative infinity. In the case of $f(x) = 4\left(\frac{4}{5}\right)^x$, since $a = 4$ and $0 < \frac{4}{5} < 1$, the function approaches $0$ as $x$ goes to positive infinity, hence the horizontal asymptote is $y = 0$.
Vertical Asymptote: Exponential functions do not have vertical asymptotes because they are defined for all real numbers $x$ and do not approach infinity or negative infinity for any finite value of $x$.
Oblique Asymptote: Exponential functions do not have oblique asymptotes. Oblique asymptotes occur when the degree of the numerator is exactly one greater than the degree of the denominator in a rational function, which is not the case for exponential functions.
Behavior of the Function: For $f(x) = 4\left(\frac{4}{5}\right)^x$, as $x$ increases, the value of the function decreases because $\frac{4}{5}$ is less than $1$. As $x$ decreases, the value of the function increases without bound.
Understanding the behavior of exponential functions and their asymptotes is crucial for graphing them and analyzing their limits as $x$ approaches positive or negative infinity.
