Graph functions f and g in the same rectangular coordinate system. Graph and give equations of all asymptotes. If applicable, use a graphing utility to confirm your hand-drawn graphs. f(x) = 3^x and g(x) = (1/3). 3^x

Verified step by step guidance

1. Identify the given functions: \(f(x) = 3^x\) and \(g(x) = \left(\frac{1}{3}\right) \cdot 3^x\).
2. Rewrite \(g(x)\) to understand its form better: since \(g(x) = \frac{1}{3} \times 3^x\), it can be expressed as \(g(x) = 3^{x-1}\) by using the property \(a^m \times a^n = a^{m+n}\).
3. Determine the domain and range of both functions: both \(f(x)\) and \(g(x)\) are exponential functions with base 3, so their domain is all real numbers \((-\infty, \infty)\) and their range is \((0, \infty)\).
4. Find the asymptotes: for both functions, the horizontal asymptote is the line \(y = 0\) because as \(x \to -\infty\), \(3^x \to 0\) and similarly for \(g(x)\).
5. Sketch the graphs on the same coordinate system: plot key points such as \(x=0\) where \(f(0) = 1\) and \(g(0) = \frac{1}{3}\), and note that \(g(x)\) is a shifted version of \(f(x)\), shifted one unit to the right. Confirm the shape and asymptotes using a graphing utility if available.

Key Concepts

Exponential Functions

Exponential functions have the form f(x) = a^x, where the base a is a positive constant. They exhibit rapid growth or decay depending on whether a is greater than or less than 1. Understanding their shape and behavior is essential for graphing and comparing functions like f(x) = 3^x and g(x) = (1/3) * 3^x.

Asymptotes of Exponential Functions

An asymptote is a line that a graph approaches but never touches. For exponential functions like f(x) = 3^x, the horizontal asymptote is typically y = 0, since the function approaches zero as x approaches negative infinity. Identifying asymptotes helps in accurately sketching the graph.

Graphing Multiple Functions on the Same Coordinate System

Plotting multiple functions together requires understanding their relative positions and transformations. For example, g(x) = (1/3) * 3^x is a vertical scaling of f(x) = 3^x by a factor of 1/3. Comparing their graphs helps visualize differences in growth rates and intercepts.