Questions: Consider the function f(x)=(1/6)^x a. Graph the function. b. Equation of the asymptote: c. Domain in interval notation: d. Range in interval notation:

Solution Steps

The horizontal asymptote of an exponential function of the form $f(x) = ab^x + c$ is $y = c$. In this case, $f(x) = (\frac{1}{6})^x$ which can be rewritten as $f(x) = 1(\frac{1}{6})^x + 0$. Therefore the horizontal asymptote is $y = 0$.

We can find two points by substituting $x$ values into the function. When $x=0$, $f(0) = (\frac{1}{6})^0 = 1$. So one point is $(0, 1)$. When $x=-1$, $f(-1) = (\frac{1}{6})^{-1} = 6$. So another point is $(-1, 6)$.

Exponential functions of the form $f(x) = ab^x$ have a domain of all real numbers. Therefore, the domain in interval notation is $(-\infty, \infty)$.

Since the horizontal asymptote is $y=0$ and the base $b = \frac{1}{6}$ is between 0 and 1, the function is decreasing and the range is all positive real numbers, excluding 0. In interval notation, the range is $(0, \infty)$.

Final Answer: