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) = (½)^x and g(x) = (1/2)^(x-1) + 1

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 whether 'a' is greater than or less than 1. In the given question, both f(x) = (½)^x and g(x) = (1/2)^(x-1) + 1 are exponential functions, with f(x) representing decay as 'a' is less than 1.

Asymptotes

Asymptotes are lines that a graph approaches but never touches or crosses. They can be vertical, horizontal, or oblique. For the functions in the question, horizontal asymptotes indicate the value that the function approaches as x approaches infinity or negative infinity. Identifying these asymptotes is crucial for understanding the behavior of the graphs of f and g.

Graphing Techniques

Graphing techniques involve plotting points and understanding the shape and behavior of functions on a coordinate system. For exponential functions, key points include the y-intercept and behavior as x approaches positive or negative infinity. Using a graphing utility can help confirm the accuracy of hand-drawn graphs, ensuring that the characteristics of the functions, including asymptotes, are correctly represented.