The figure shows the graph of f(x) = e^x. In Exercises 35-46, use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine graphs. each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn g(x) = e^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 case where the base 'e' (approximately 2.718) is used. 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 shifting, stretching, compressing, or reflecting the graph of a function. For example, adding a constant to the function, as in g(x) = e^x + 2, results in a vertical shift of the graph upward by 2 units. Understanding these transformations is essential for accurately graphing modified functions based on their parent functions.

Asymptotes

Asymptotes are lines that a graph approaches but never touches or crosses. For exponential functions like f(x) = e^x, the horizontal asymptote is typically at y = 0, indicating that as x approaches negative infinity, the function approaches this line. Identifying asymptotes is crucial for understanding the behavior of the graph, particularly in determining the domain and range of the function.