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A linear function is a function that forms a straight line when graphed. These functions are among the simplest and most fundamental in algebra. Typically, a linear function is written in the form \(f(x) = mx + b\). Here, \(m\) represents the slope and \(b\) is the y-intercept.

The slope signifies the rate of change of the function. It tells you how steep the line is. If you have the slope \(\frac{2}{9}\), it means that for every 9 units you move right, the line goes up by 2 units. The slope is essential because it provides the direction and tilt of the line.

The y-intercept, on the other hand, is the point where the line crosses the y-axis. For example, in the function \(f(x) = \frac{2}{9}x + 4\), the y-intercept is 4. It's where the function will always start when you plot it on a graph. Understanding these components makes graphing and analyzing linear functions much easier.

Graphing functions involves plotting points on a grid to visualize the relationship between variables—a core part of understanding functions in mathematics.

When graphing the linear function \(f(x) = \frac{2}{9}x + 4\), start by marking the y-intercept at 4 on the y-axis. Next, use the slope \(\frac{2}{9}\) to find another point. From (0, 4), move up 2 and right 9 to point (9, 6). Draw a straight line through these points. This line represents the entire function graphically.

For the inverse function \(f^{-1}(x) = \frac{9}{2}x - 18\), begin by plotting the y-intercept at -18. Use the slope of \(\frac{9}{2}\) to identify the point (2, -9). Move upward 9 and right 2 from the y-intercept. Connect these points to form a line. This line is the inverse of the original function and mirrors it with respect to the line \(y = x\).

Graphing these functions helps in examining how the original and the inverse functions relate visually.

Composing functions is joining two functions together to form a new function. When dealing with inverse functions, checking if two functions truly are inverses is essential.

To determine if \(f(x)\) and \(f^{-1}(x)\) are inverses, create compositions: \(f(f^{-1}(x))\) and \(f^{-1}(f(x))\). If both equal \(x\), the functions are inverses.

Consider \(f(f^{-1}(x))\). Substitute \(f^{-1}(x) = \frac{9}{2}x - 18\) into \(f(x)\):
\[f\left(\frac{9}{2}x - 18\right) = \frac{2}{9} \left(\frac{9}{2}x - 18\right) + 4 = x.\]
This correctly simplifies to \(x\), testifying that the input and output are identical.

Next, verify with \(f^{-1}(f(x))\). Plug \(f(x) = \frac{2}{9}x + 4\) into \(f^{-1}(x)\):
\[f^{-1}\left(\frac{2}{9}x + 4\right) = \frac{9}{2}\left(\frac{2}{9}x + 4\right) - 18 = x.\]
This also equals \(x\), completing the verification. Therefore, both compositions resulting in \(x\) confirms their inverse relationship.