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-1

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.

Transformations of Functions

Transformations of functions involve shifting, reflecting, stretching, or compressing the graph of a function. Common transformations include vertical shifts (adding or subtracting a constant), horizontal shifts (adding or subtracting from the input), and reflections (flipping the graph over an axis). For the function g(x) = e^(x) - 1, the graph of f(x) = e^x is shifted down by 1 unit, affecting its asymptote and range.

Asymptotes

Asymptotes are lines that a graph approaches but never touches or crosses. For exponential functions, the horizontal asymptote is typically found at y = k, where k is a constant that represents the vertical shift of the function. In the case of g(x) = e^x - 1, the horizontal asymptote is y = -1, indicating that as x approaches negative infinity, the function approaches this line.