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

Verified step by step guidance

1. 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.
2. 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.
3. 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.
4. 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 \(x > 0\).
5. 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 \(y = x\).

Key Concepts

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.

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.

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.