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) = 2.2^x

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 horizontal asymptote at \(y = 0\), domain \((-\infty, \infty)\), and range \((0, \infty)\).
2. Next, analyze the given function \(g(x) = 2 \cdot 2^x\). Notice that this can be rewritten as \(g(x) = 2^{x+1}\) because \$2 \cdot 2^x = 2^{1} \cdot 2^x = 2^{x+1}$.
3. Recognize that \(g(x) = 2^{x+1}\) represents a horizontal shift of the base graph \(f(x) = 2^x\) to the left by 1 unit. This is because adding 1 inside the exponent shifts the graph left.
4. The horizontal asymptote remains the same at \(y = 0\) since multiplying by a positive constant does not change the asymptote for exponential functions with positive bases.
5. Finally, determine the domain and range of \(g(x)\). The domain remains all real numbers \((-\infty, \infty)\), and the range remains \((0, \infty)\) because the function is still an exponential growth function shifted horizontally.

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 a horizontal asymptote. Understanding the basic graph of f(x) = 2^x is essential before applying transformations.

Transformations of Functions

Transformations involve shifting, stretching, compressing, or reflecting the graph of a function. For g(x) = 2·2^x, the coefficient 2 vertically stretches the graph of f(x) = 2^x by a factor of 2. 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 and stretches. The horizontal asymptote, often y=0 for basic exponentials, may shift with transformations. Identifying these features is crucial for understanding the behavior and limits of the function.