Here are the essential concepts you must grasp in order to answer the question correctly.
Exponential Functions
Exponential functions are mathematical expressions in the form f(x) = a^x, where 'a' is a positive constant. In this case, f(x) = (1/2)^x represents a decreasing exponential function, as the base (1/2) is less than 1. Understanding the behavior of exponential functions is crucial for graphing them, as they exhibit rapid growth or decay depending on the base.
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Logarithmic Functions
Logarithmic functions are the inverses of exponential functions and are expressed as g(x) = log_b(x), where 'b' is the base. For g(x) = log(1/2)(x), the base is 1/2, indicating that this function will decrease as x increases. Recognizing the properties of logarithms, such as their domain and range, is essential for accurately graphing them alongside exponential functions.
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Graphs of Logarithmic Functions
Graphing in Rectangular Coordinate Systems
Graphing in rectangular coordinate systems involves plotting points on a two-dimensional plane defined by an x-axis and a y-axis. Each function's graph is represented by a set of points (x, f(x)) or (x, g(x)). Understanding how to scale the axes and interpret the intersection points of different functions is vital for visualizing their relationships and behaviors in the same coordinate system.
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Graphs & the Rectangular Coordinate System