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

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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.

Identify the domain of the function. Since \$3^x$ is an exponential function defined for all real numbers, the domain is $(-\infty, \infty)$.

Determine the range of the function. Because \$3^x$ is always positive for any real $x$, the range is $(0, \infty)$.

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.

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.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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