In Exercises 25-34, 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

Verified step by step guidance

1. Graph the basic function $f(x) = 2^x$. This is an exponential function with a base greater than 1, so it will be an increasing curve that passes through the point (0,1) and approaches the x-axis as an asymptote as x approaches negative infinity.
2. Identify the horizontal asymptote of $f(x) = 2^x$. Since the function approaches the x-axis but never touches it, the horizontal asymptote is $y = 0$.
3. Apply the transformation to graph $g(x) = -2^x$. This transformation involves reflecting the graph of $f(x) = 2^x$ across the x-axis. This means that for every point (x, y) on the graph of $f(x)$, there will be a corresponding point (x, -y) on the graph of $g(x)$.
4. Determine the asymptote for $g(x) = -2^x$. The reflection across the x-axis does not change the horizontal asymptote, so it remains $y = 0$.
5. Determine the domain and range of $g(x) = -2^x$. The domain of an exponential function is all real numbers, so the domain is $(-\infty, \infty)$. The range, after reflection, is $(-\infty, 0)$, as the function now takes on negative values only.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in the form f(x) = a^x, where 'a' is a positive constant. They exhibit rapid growth or decay depending on the base value. Understanding the basic shape and behavior of the graph of f(x) = 2^x is crucial, as it serves as the foundation for applying transformations to graph related functions.

Transformations of Functions

Transformations involve altering the graph of a function through shifts, stretches, compressions, or reflections. For example, the function g(x) = -2^x represents a vertical reflection of f(x) = 2^x across the x-axis. Recognizing how these transformations affect the graph's position and shape is essential for accurately graphing the new function.

Asymptotes

Asymptotes are lines that a graph approaches but never touches. For exponential functions, the horizontal asymptote is typically the x-axis (y = 0). Identifying asymptotes helps in understanding the behavior of the function as x approaches positive or negative infinity, which is important for determining the domain and range of the function.