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 a horizontal asymptote at \(y = 0\), domain \((-\infty, \infty)\), and range \((0, \infty)\).
2. Identify the transformation in \(g(x) = 2^{x+1}\). The expression \(x + 1\) inside the exponent indicates a horizontal shift of the graph of \(f(x)\) to the left by 1 unit.
3. Graph the transformed function \(g(x)\) by shifting every point on the graph of \(f(x)\) one unit to the left. For example, the point \((0, 1)\) on \(f(x)\) moves to \((-1, 1)\) on \(g(x)\).
4. Determine the equation of the asymptote for \(g(x)\). Since the transformation is horizontal, the horizontal asymptote remains \(y = 0\).
5. Use the graph to state the domain and range of \(g(x)\). The domain remains all real numbers \((-\infty, \infty)\), and the range remains \((0, \infty)\) because the transformation does not affect these.

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 before applying transformations.

Transformations of Functions

Transformations modify the graph of a base function by shifting, stretching, compressing, or reflecting it. For g(x) = 2^(x+1), the '+1' inside the exponent shifts the graph horizontally to the left by 1 unit. Recognizing how changes inside the function affect the graph helps in sketching and interpreting new functions.

Asymptotes, Domain, and Range

An asymptote is a line that the graph approaches but never touches; for exponential functions like 2^x, the horizontal asymptote is y=0. The domain of exponential functions is all real numbers, while the range is positive real numbers (y > 0). Identifying asymptotes and determining domain and range are key to fully understanding the function's behavior.