Graph each function. Give the domain and range. See Example 3. ƒ(x) = (1/3)^-x

Verified step by step guidance

1. Rewrite the function to a more familiar form by using the property of exponents: \(f(x) = \left(\frac{1}{3}\right)^{-x} = 3^x\). This will make it easier to analyze and graph.
2. Identify the domain of the function. Since \$3^x$ is an exponential function defined for all real numbers, the domain is \((-\infty, \infty)\).
3. Determine the range of the function. Because \$3^x\( is always positive for any real \)x$, the range is \((0, \infty)\).
4. Plot key points to help graph the function, such as \(f(0) = 3^0 = 1\), \(f(1) = 3^1 = 3\), and \(f(-1) = 3^{-1} = \frac{1}{3}\). These points show the exponential growth behavior.
5. Sketch the graph using the points and the knowledge that the function approaches zero but never touches the x-axis (horizontal asymptote at \(y=0\)), and increases rapidly as \(x\) increases.

Key Concepts

Exponential Functions

An exponential function has the form f(x) = a^x, where the base a is a positive real number not equal to 1. In this problem, the function involves a negative exponent, which affects the graph by reflecting it across the y-axis. Understanding how the base and exponent influence the shape is essential for graphing.

Domain and Range of Functions

The domain is the set of all possible input values (x-values) for the function, while the range is the set of all possible output values (f(x)). For exponential functions like f(x) = (1/3)^-x, the domain is all real numbers, and the range is all positive real numbers, since exponential functions never output zero or negative values.

Negative Exponents and Their Properties

A negative exponent indicates the reciprocal of the base raised to the positive exponent, i.e., a^{-x} = 1 / a^x. This property transforms the function and affects its graph, often reflecting it or changing its growth behavior. Recognizing this helps in rewriting and graphing the function accurately.