In Exercises 5–9, graph f and g in the same rectangular coordinate system. Use transformations of the graph of f to obtain the graph of g. Graph and give equations of all asymptotes. Use the graphs to determine each function's domain and range. f(x) = e^x and g(x) = 2e^(x/2)

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in the form f(x) = a * b^x, where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent. The function f(x) = e^x is a specific exponential function where 'e' is Euler's number, approximately 2.718. These functions are characterized by their rapid growth or decay and have unique properties such as a horizontal asymptote at y = 0.

Transformations of Functions

Transformations of functions involve altering the graph of a parent function to create a new function. Common transformations include vertical and horizontal shifts, reflections, and stretches or compressions. For example, the function g(x) = 2e^(x/2) can be derived from f(x) = e^x by applying a horizontal compression (due to the x/2) and a vertical stretch (due to the factor of 2). Understanding these transformations is crucial for graphing and analyzing functions.

Asymptotes

Asymptotes are lines that a graph approaches but never touches or crosses. For exponential functions, the horizontal asymptote is typically found at y = 0, indicating that as x approaches negative infinity, the function approaches this line. Identifying asymptotes is essential for understanding the behavior of functions at extreme values and helps in determining the domain and range of the functions involved.