Find the Asymptotes f(x)=2(2/5)^x

Solution:

Step 1:

Identify the horizontal asymptote for the given exponential function. Since exponential functions approach a constant value as $x$ goes to positive or negative infinity, the horizontal asymptote is the value the function approaches. For $f(x)=2(2/5{)}^x$, as $x$ increases or decreases indefinitely, the function approaches $y=0$. Therefore, the horizontal asymptote is given by the equation $y=0$.

Step 2:

There are no vertical asymptotes for this function. 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 $x$.

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 properties:

1. Horizontal Asymptote: The horizontal asymptote of an exponential function is determined by the constant term $a$. If $a$ is zero, the horizontal asymptote is $y=0$. If $a$ is not zero, and $b$ is between 0 and 1, the horizontal asymptote is still $y=0$ because as $x$ goes to infinity, $b^x$ goes to zero, and the function approaches the value of $a$. In the case of $f(x)=2(2/5{)}^x$, the constant term is 2, and the base $(2/5)$ is between 0 and 1, so as $x$ goes to positive or negative infinity, $f(x)$ approaches 0.

2. Vertical Asymptote: Exponential functions do not have vertical asymptotes. This is because the function is defined for all real numbers $x$ and does not tend towards infinity or negative infinity at any finite value of $x$.

3. Domain and Range: The domain of an exponential function is all real numbers, $(-\mathrm{\infty },\mathrm{\infty })$. The range is $(0,\mathrm{\infty })$ if $a$ is positive and $(-\mathrm{\infty },0)$ if $a$ is negative, assuming $b>1$. For $0<b<1$, the range is reversed.

4. Growth and Decay: If $b>1$, the function represents exponential growth; if $0<b<1$, it represents exponential decay. In the given problem, $b=2/5$, which is less than 1, indicating exponential decay.

Understanding these properties helps in determining the behavior of the function and in identifying any asymptotes.