In Exercises 5–9, graph f and g in the same rectangular coordinate system. Use transformations of the graph of f to obtain the graph of g. Graph and give equations of all asymptotes. Use the graphs to determine each function's domain and range. f(x) = 3^x and g(x) = -3^x

Verified step by step guidance

1. Identify the base function and its transformation: The base function is \(f(x) = 3^x\), which is an exponential function with base 3.
2. Understand the transformation from \(f(x)\) to \(g(x)\): The function \(g(x) = -3^x\) is obtained by reflecting the graph of \(f(x)\) across the x-axis because of the negative sign in front.
3. Graph \(f(x) = 3^x\): Plot points for several values of \(x\) (e.g., \(x = -2, -1, 0, 1, 2\)) to see the exponential growth. Note that \(f(x)\) approaches 0 as \(x\) approaches negative infinity but never touches the x-axis.
4. Graph \(g(x) = -3^x\): Use the reflection transformation on the points of \(f(x)\) by multiplying the \(y\)-values by -1. This flips the graph over the x-axis, so it approaches 0 from below as \(x\) approaches negative infinity.
5. Determine the asymptotes, domain, and range: Both functions have a horizontal asymptote at \(y=0\). The domain for both is all real numbers, \((-\infty, \infty)\). The range of \(f(x)\) is \((0, \infty)\), and the range of \(g(x)\) is \((-\infty, 0)\).

Key Concepts

Exponential Functions and Their Graphs

An exponential function has the form f(x) = a^x, where a > 0 and a ≠ 1. Its graph shows rapid growth or decay depending on the base. Understanding the shape and behavior of f(x) = 3^x is essential for graphing and comparing transformations.

Transformations of Functions

Transformations include shifts, reflections, stretches, and compressions of a graph. For g(x) = -3^x, the negative sign reflects the graph of f(x) = 3^x across the x-axis. Recognizing how these changes affect the graph helps in sketching g from f.

Asymptotes, Domain, and Range

An asymptote is a line that the graph approaches but never touches. For exponential functions like f(x) = 3^x, the horizontal asymptote is y = 0. Identifying asymptotes aids in determining the domain (all real x) and range (values y can take) of the functions.