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) = 3^-x

Verified step by step guidance

1. Identify the functions given: \(f(x) = 3^x\) and \(g(x) = 3^{-x}\). These are exponential functions with base 3, where \(f(x)\) is increasing and \(g(x)\) is decreasing.
2. Determine the domain and range of both functions. Both \(f(x)\) and \(g(x)\) have domain \((-\infty, \infty)\) and range \((0, \infty)\) because exponential functions with positive bases are always positive.
3. Find the asymptotes for both functions. Since exponential functions approach zero but never touch it, the horizontal asymptote for both \(f(x)\) and \(g(x)\) is \(y = 0\).
4. Sketch the graphs on the same coordinate system: For \(f(x) = 3^x\), plot points for several values of \(x\) (e.g., \(x = -1, 0, 1\)) to see the exponential growth. For \(g(x) = 3^{-x}\), plot points similarly to observe the exponential decay.
5. Use a graphing utility to confirm your hand-drawn graphs and verify the horizontal asymptote \(y = 0\) for both functions. Note how \(f(x)\) increases rapidly as \(x\) increases, while \(g(x)\) decreases towards zero as \(x\) increases.

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 the exponent is positive or negative. Understanding their general shape and behavior is essential for graphing and comparing functions like f(x) = 3^x and g(x) = 3^-x.

Asymptotes of Exponential Functions

An asymptote is a line that a graph approaches but never touches. For exponential functions with positive bases, the horizontal asymptote is typically the x-axis (y=0). Recognizing and writing the equations of asymptotes helps in accurately sketching the graph and understanding the function's long-term behavior.

Graphing and Using Graphing Utilities

Graphing functions by hand involves plotting key points and understanding function behavior, while graphing utilities provide precise visualizations. Using both methods allows verification of the graph's accuracy and helps identify features like intercepts and asymptotes, enhancing comprehension of the functions' properties.