how do the graphs of the functions $f(x)=(\frac{3}{2})^x$ and $g(x)=(\frac{2}{3})^x$ compare?
how do the graphs of the functions $f(x)=(\frac{3}{2})^x$ and $g(x)=(\frac{2}{3})^x$ compare?
Answer
Explanation:
Step1: Recall exponential - function properties
For an exponential function (y = a^x), if (a>1), the function is increasing, and if (0 < a<1), the function is decreasing. For (f(x)=\left(\frac{3}{2}\right)^x), since (a = \frac{3}{2}>1), (y = f(x)) is an increasing exponential function. Its (y) - values increase as (x) increases. The graph passes through the point ((0,1)) (because when (x = 0), (f(0)=\left(\frac{3}{2}\right)^0=1)) and goes to (+\infty) as (x\rightarrow+\infty) and approaches (0) as (x\rightarrow-\infty). For (g(x)=\left(\frac{2}{3}\right)^x), since (0<\frac{2}{3}<1), (y = g(x)) is a decreasing exponential function. Its (y) - values decrease as (x) increases. The graph also passes through the point ((0,1)) (because when (x = 0), (g(0)=\left(\frac{2}{3}\right)^0 = 1)), goes to (+\infty) as (x\rightarrow-\infty) and approaches (0) as (x\rightarrow+\infty).
Step2: Analyze the relationship between the two graphs
The two functions (f(x)=\left(\frac{3}{2}\right)^x) and (g(x)=\left(\frac{2}{3}\right)^x) are related in that (g(x)=\frac{1}{f(x)}). The graph of (y = g(x)) is the reflection of the graph of (y = f(x)) across the (y) - axis. This is because if we replace (x) with (-x) in (f(x)), we get (f(-x)=\left(\frac{3}{2}\right)^{-x}=\left(\frac{2}{3}\right)^x=g(x)).
Answer:
The graph of (f(x)=\left(\frac{3}{2}\right)^x) is an increasing exponential function, and the graph of (g(x)=\left(\frac{2}{3}\right)^x) is a decreasing exponential function. The graph of (g(x)) is the reflection of the graph of (f(x)) across the (y) - axis.