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) = (1/3). 3^x

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'. For example, f(x) = 3^x grows quickly as x increases, while g(x) = (1/3) * 3^x represents a scaled version of the same exponential function, affecting its amplitude but not its growth behavior.

Asymptotes are lines that a graph approaches but never touches. They can be vertical, horizontal, or oblique. For exponential functions, horizontal asymptotes often occur at y = 0, indicating that as x approaches negative infinity, the function values approach zero. Understanding asymptotes is crucial for accurately graphing functions and predicting their behavior at extreme values.

Graphing utilities are software tools or calculators that allow users to visualize mathematical functions. They can plot graphs, find intersections, and confirm hand-drawn sketches. Using a graphing utility can help verify the accuracy of asymptotes and the overall shape of the functions, providing a clearer understanding of their behavior across different values of x.