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Function notation is a way to represent functions in mathematics using symbols. It helps in simplifying and better understanding of complex expressions. Instead of writing an equation like "y equals x squared," you can use function notation such as \(f(x) = x^2\). This makes it clear that \(f\) is a function that takes \(x\) as its input and returns \(x^2\) as its output.

In function notation, you might come across expressions like \((f \circ g)(x)\). This denotes the composition of the two functions \(f\) and \(g\), which means you first apply \(g\) to \(x\) and then apply \(f\) to the result of \(g(x)\). Function notation allows us to easily combine functions and observe the step-by-step effects each function has on an input value.

Quadratic functions belong to a family of functions characterized by their characteristic U-shaped curve called a parabola. The standard form of a quadratic function is \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants with \(a eq 0\).

If we focus on \(f(x) = x^2\), like in our exercise, it is a simple quadratic function where \(a = 1\), and \(b\) and \(c\) are zero. This particular kind of quadratic function is symmetric around the y-axis, and its graph opens upwards. Quadratic functions are widely utilized in various fields, including physics and engineering, due to their predictable properties and ability to model different types of phenomena.

Substitution is a powerful tool in mathematics used to solve expressions or systems of equations by replacing variables with their corresponding values or expressions. It simplifies and reduces problems to a form that can be easily solved.

In the process we discussed earlier, the key to solving \((f \circ g)(-a)\) is substitution. First, evaluate \(g(-a)\) by substituting \(-a\) into the function \(g(x) = x - 3\). This yields \(g(-a) = -(a + 3)\).

Next, we substitute the result of \(g(-a)\) into \(f(x)\). By replacing \(x\) in \(f(x)\) with \(-(a + 3)\), we get \(f(-(a+3)) = [-(a + 3)]^2\). By substituting and simplifying using the exponent rule, we achieve the final expression \((a + 3)^2\). Substitution thus transforms complex expressions into simpler, solvable ones.