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Composite functions are powerful tools in mathematics that allow us to apply one function to the results of another. This is essentially evaluating a function at the output of another function. In simpler terms, you're stacking one function on top of another.

For example, if you have two functions, say \(f(x)\) and \(g(x)\), you might want to find \(f(g(x))\). This expression represents the composite function where the result of \(g(x)\) becomes the input of \(f(x)\).

Evaluating composite functions involves two major steps:

• First, find the output of the first function using the given input.
• Next, take this output and use it as the input for the second function.

In the exercise provided, we are tasked with finding \((fg)(-3)\), which requires us to evaluate \(g(x)\) at \(-3\) and then use this result in \(f(x)\). This example clearly shows how composite functions work in a practical scenario.

Quadratic functions are some of the most common polynomial functions you'll encounter in algebra. The general form of a quadratic function is \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\).

This type of function graphs as a parabola on the coordinate plane. The direction of the parabola (whether it opens upwards or downwards) depends on the sign of \(a\):

• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), the parabola opens downwards.

The vertex of the parabola is the highest or lowest point depending on the direction the parabola opens. This is also a point of symmetry for the function.

In our exercise, the function \(f(x) = x^2 - 3x + 2\) is quadratic. Evaluating this function is just a matter of substituting the respective input value and simplifying.

Linear functions are the simplest type of function, characterized by the property that their graph is a straight line. The standard form of a linear function is \(g(x) = mx + b\), where \(m\) represents the slope and \(b\) the y-intercept.

The slope \(m\) indicates the steepness and direction of the line:

• A positive slope means the line rises as it moves from left to right.
• A negative slope means the line falls as it moves from left to right.

Meanwhile, the y-intercept \(b\) is the point where the line crosses the y-axis.

For the exercise, the function \(g(x) = 2x - 4\) is a linear function. This means for any input \(x\), the output is simply determined by the linear equation. Calculating lines like this involves simple substitution and arithmetic for any given \(x\) value.