In Exercises 47–52, graph functions f and g in the same rectangular coordinate system. Graph and give equations of all asymptotes. If applicable, use a graphing utility to confirm your hand-drawn graphs. f(x) = 3^x and g(x) = 3^-x

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in the form f(x) = a^x, where 'a' is a positive constant. These functions exhibit rapid growth or decay depending on the base 'a'. In this case, f(x) = 3^x represents exponential growth, while g(x) = 3^-x represents exponential decay, as the negative exponent indicates a reciprocal relationship.

Asymptotes

Asymptotes are lines that a graph approaches but never touches. They can be vertical, horizontal, or oblique. For the functions f(x) = 3^x and g(x) = 3^-x, the horizontal asymptote is y = 0, indicating that as x approaches negative infinity, the function values approach zero but do not reach it.

Graphing Techniques

Graphing techniques involve plotting points and understanding the behavior of functions to create accurate visual representations. For the given functions, identifying key points, such as intercepts and asymptotes, helps in sketching the graphs. Using a graphing utility can provide confirmation and a more precise depiction of the functions' behavior across different values of x.