Find the Asymptotes f(x)=4(4/5)^x

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 $f(x)=a{b}^x$, where $a$ is a constant, $b$ is the base of the exponential (with $b>0$ and $b\ne 1$), and $x$ is the exponent. These functions have the following characteristics:

1. 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 $y=0$.

2. 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 $x$.

3. 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.

4. 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.