Graph f(x) = (1/2)^x and g(x) = log_1/2 x in the same rectangular coordinate system.

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Identify the functions to be graphed: \(f(x) = \left(\frac{1}{2}\right)^x\) is an exponential function with base \(\frac{1}{2}\), and \(g(x) = \log_{\frac{1}{2}}(x)\) is the logarithmic function with base \(\frac{1}{2}\).

Recall that the graph of \(f(x) = a^x\) where \(0 < a < 1\) is a decreasing exponential curve that passes through the point \((0,1)\) because any nonzero number raised to the zero power is 1.

For \(g(x) = \log_a(x)\) where \(0 < a < 1\), the graph is the inverse of the exponential function \(f(x) = a^x\). This means the graph of \(g(x)\) is a reflection of \(f(x)\) across the line \(y = x\).

Determine key points for \(f(x)\) by substituting values such as \(x = -1, 0, 1, 2\) to get points like \((x, f(x))\). Similarly, find points for \(g(x)\) by choosing \(x\) values and calculating \(g(x)\), keeping in mind the domain of \(g(x)\) is \(x > 0\).

Plot the points for both functions on the same coordinate system, draw smooth curves through these points, and include the line \(y = x\) to visualize the reflection relationship between \(f(x)\) and \(g(x)\).

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Exponential Functions

An exponential function has the form f(x) = a^x, where the base a is a positive constant. In this question, f(x) = (1/2)^x represents exponential decay since the base is between 0 and 1. Understanding how the graph behaves, including its asymptote and decreasing nature, is essential for plotting it accurately.

Logarithmic Functions

A logarithmic function is the inverse of an exponential function and is written as g(x) = log_a(x), where a is the base. Here, g(x) = log_(1/2)(x) is the logarithm with base 1/2, which is less than 1, causing the graph to decrease and reflect the inverse behavior of the exponential function. Recognizing domain restrictions and shape is key.

Inverse Functions and Their Graphs

Exponential and logarithmic functions with the same base are inverses, meaning their graphs are reflections across the line y = x. Understanding this relationship helps in graphing both functions on the same coordinate system and predicting their intersection and symmetry properties.

Graph f(x) = (1/2)x and g(x) = log1/2 x in the same rectangular coordinate system.