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 – 1

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 obtain \( g(x) = 2^x - 1 \). This transformation is a vertical shift downward by 1 unit. Therefore, every point on the graph of \( f(x) = 2^x \) is moved down by 1 unit.
4. Determine the new horizontal asymptote for \( g(x) = 2^x - 1 \). Since the entire graph is shifted down by 1 unit, the horizontal asymptote is now \( y = -1 \).
5. Use the graph to determine the domain and range of \( g(x) = 2^x - 1 \). The domain remains all real numbers \( (-\infty, \infty) \), and the range is all real numbers greater than -1, so \( (-1, \infty) \).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in the form f(x) = a^x, where 'a' is a positive constant. They are characterized by their rapid growth or decay, depending on the base. The function f(x) = 2^x, for example, increases exponentially as x increases. Understanding the properties of exponential functions is crucial for graphing and analyzing their behavior.

Transformations of Functions

Transformations of functions involve shifting, stretching, compressing, or reflecting the graph of a function. For instance, the function g(x) = 2^x - 1 represents a vertical shift of the graph of f(x) = 2^x downward by one unit. Recognizing how these transformations affect the original graph is essential for accurately graphing new functions and understanding their characteristics.

Asymptotes

Asymptotes are lines that a graph approaches but never touches. For exponential functions, the horizontal asymptote is often found at y = k, where k is a constant. In the case of g(x) = 2^x - 1, the horizontal asymptote is y = -1. Identifying asymptotes helps in determining the behavior of the function as x approaches positive or negative infinity, which is vital for understanding the domain and range.