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)

Verified step by step guidance

1. Identify the horizontal asymptote of the graph, which is y = -1.
2. Recognize that the horizontal asymptote y = -1 suggests a vertical shift of the function downward by 1 unit.
3. Consider the function h(x) = 3^x - 1, which has a vertical shift downward by 1 unit.
4. Check the behavior of the graph: it is increasing and passes through the point (0, 0), which matches the function h(x) = 3^x - 1.
5. Conclude that the graph represents the function h(x) = 3^x - 1 based on the asymptote and the increasing nature of the graph.

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. These functions exhibit rapid growth or decay, depending on the base. For example, f(x) = 3^x grows quickly as x increases, while g(x) = 3^(-x) represents exponential decay.

Horizontal Asymptotes

A horizontal asymptote is a horizontal line that a graph approaches as x approaches positive or negative infinity. In the context of exponential functions, if the function approaches a constant value as x goes to infinity or negative infinity, that constant is the horizontal asymptote. In the given graph, the horizontal asymptote is y = -1, indicating that the function approaches this line but never actually reaches it.

Transformations of Functions

Transformations of functions involve shifting, reflecting, stretching, or compressing the graph of a function. For example, the function g(x) = 3^(x-1) represents a horizontal shift of the base function f(x) = 3^x to the right by 1 unit. Similarly, h(x) = 3^x - 1 indicates a vertical shift downward by 1 unit. Understanding these transformations is crucial for identifying the correct function based on its graph.