Begin by graphing f(x) = 2^x. Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs. g(x) = 2^x – 1

Verified step by step guidance

1. Start by understanding the base function: \(f(x) = 2^{x}\). This is an exponential function with base 2, which has a graph that passes through the point \((0,1)\) and increases rapidly as \(x\) increases. The horizontal asymptote of this graph is the line \(y = 0\).
2. Next, analyze the given function \(g(x) = 2^{x} - 1\). Notice that this is a vertical shift of the base function \(f(x)\) downward by 1 unit because of the \(-1\) outside the exponential.
3. To graph \(g(x)\), take the graph of \(f(x) = 2^{x}\) and shift every point down by 1 unit. For example, the point \((0,1)\) on \(f(x)\) moves to \((0, 0)\) on \(g(x)\).
4. Determine the new horizontal asymptote for \(g(x)\). Since the original asymptote was \(y = 0\), shifting down by 1 moves the asymptote to \(y = -1\). This means the graph will approach but never cross the line \(y = -1\).
5. Finally, use the graph to identify the domain and range of \(g(x)\). The domain of an exponential function is all real numbers, so \((-\infty, \infty)\). The range is all values greater than the horizontal asymptote, so for \(g(x)\) it is \((-1, \infty)\).

Key Concepts

Exponential Functions

An exponential function has the form f(x) = a^x, where the base a is a positive constant not equal to 1. These functions grow or decay rapidly and have unique properties such as always being positive and having a horizontal asymptote. Understanding the basic graph of f(x) = 2^x is essential for applying transformations.

Transformations of Functions

Transformations involve shifting, stretching, compressing, or reflecting the graph of a function. For g(x) = 2^x - 1, subtracting 1 shifts the graph of 2^x downward by 1 unit. Recognizing how these changes affect the graph and asymptotes helps in sketching the new function accurately.

Domain, Range, and Asymptotes

The domain of exponential functions is all real numbers, while the range depends on vertical shifts. The horizontal asymptote is a line the graph approaches but never touches; for f(x) = 2^x, it is y = 0. After transformations, the asymptote shifts accordingly, affecting the range and graph behavior.