Graph f(x) = 4x and g(x) = log4 x in the same rectangular coordinate system.

Ch. 4 - Exponential and Logarithmic Functions

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Ch. 4 - Exponential and Logarithmic Functions

Problema 43

Capitolo 5, Problema 43

Graph f(x) = 4^x and g(x) = log₄ x in the same rectangular coordinate system.

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Recognize that the functions given are inverses of each other: \(f(x) = 4^x\) is an exponential function, and \(g(x) = \log_4 x\) is its inverse logarithmic function.

Create a table of values for \(f(x) = 4^x\) by choosing several values of \(x\) (such as \(-2\), \(-1\), \(0\), \(1\), \(2\)) and calculating the corresponding \(f(x)\) values.

Create a table of values for \(g(x) = \log_4 x\) by choosing several positive values of \(x\) (such as \(\frac{1}{16}\), \(\frac{1}{4}\), \(1\), \(4\), \(16\)) and calculating the corresponding \(g(x)\) values.

Plot the points from both tables on the same rectangular coordinate system, noting that \(f(x)\) will be increasing and \(g(x)\) will be increasing but only defined for \(x > 0\).

Draw the graphs smoothly through the plotted points, remembering that the graph of \(g(x)\) is the reflection of the graph of \(f(x)\) across the line \(y = x\).

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Exponential Functions

An exponential function has the form f(x) = a^x, where the base a is a positive constant not equal to 1. It models rapid growth or decay and has a domain of all real numbers and a range of positive real numbers. For f(x) = 4^x, the graph passes through (0,1) and increases rapidly as x increases.

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Exponential Functions

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. It is defined only for positive x-values and has a range of all real numbers. For g(x) = log_4(x), the graph passes through (1,0) and increases slowly, reflecting the inverse relationship to 4^x.

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Graphs of Logarithmic Functions

Inverse Functions and Their Graphs

Inverse functions reverse the effect of each other, so their graphs are symmetric about the line y = x. Since g(x) = log_4(x) is the inverse of f(x) = 4^x, plotting both on the same coordinate system shows this symmetry, helping to understand their relationship visually.

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Graphs of Logarithmic Functions

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