In Exercises 19–24, the graph of an exponential function is given. Select the function for each graph from the following options: f(x) = 3^x, g(x) = 3^(x-1), h(x) = 3^x - 1 ; f(x) = -3^x, G(x) = 3^(-x), H(x) = -3^(-x)

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions of 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. These functions exhibit rapid growth or decay, depending on the base. For example, f(x) = 3^x grows quickly as x increases, while f(x) = 3^(-x) decays as x increases.

Transformations of Functions

Transformations of functions involve shifting, stretching, compressing, or reflecting the graph of a function. For instance, g(x) = 3^(x-1) represents a horizontal shift of the function f(x) = 3^x to the right by 1 unit, while h(x) = 3^x - 1 shifts it down by 1 unit. Understanding these transformations is crucial for accurately identifying the function represented by a graph.

Graph Characteristics of Exponential Functions

The graph of an exponential function typically features a horizontal asymptote, which is a line that the graph approaches but never touches. For example, the graph of f(x) = 3^x approaches the x-axis (y=0) as x approaches negative infinity. Recognizing these characteristics helps in distinguishing between different exponential functions based on their graphs.