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Composite functions occur when one function is applied and then another function is applied to the result of the first. They are commonly notated as \((f \circ g)(x)\) or \((g \circ f)(x)\). Think of it as a two-step process, where you first plug the input into the inner function \(g(x)\), and then use the output of \(g(x)\) as the input for the outer function \(f(x)\).

In the exercise, to find \(f \circ g(x)\), you substitute \(g(x)\) into \(f(x)\). This follows with \(g \circ f(x)\), where you plug \(f(x)\) into \(g(x)\). These operations reflect how functions can be interconnected, producing outputs that depend on a sequence of transformations.

• To form \(f \circ g\), we input \(x+2\) into \(f(x)\): this gives us a new quadratic function.
• For \(g \circ f\), we input \(3x^2 + 4x\) into \(g(x)\): this gives us a modified linear function.

The domain of a function is the set of all possible inputs (usually represented as 'x' values) for which the function is defined. Ensuring that the domain is correctly identified is crucial for understanding the limits within which a function operates.

For composite functions, the domain of the resultant function may be influenced by the domains of the individual functions involved. If \(g(x)\) is the inner function and \(f(x)\) is the outer function in \(f \circ g(x)\), then we need the outputs from \(g(x)\) to be within the domain of \(f(x)\) - sometimes this can restrict the domain of the composite function.

However, in our problem, since both functions involved are defined for all real numbers, the domain of \(f \circ g\) and \(g \circ f\) remains all real numbers. This makes it easier for us to understand and work with these composite functions.

Quadratic functions are polynomial functions with the highest exponent of 2, and they are typically written in the form \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. Graphically, they represent parabolas, which open upwards or downwards based on the sign of \(a\).

In our case, \(f(x) = 3x^2 + 4x\) is the quadratic function. When working with composite functions like \(f \circ g\), substituting a linear function into this quadratic alters its original shape, but it remains a parabola nonetheless.

• The coefficient \(3\) indicates that the parabola opens upwards.
• The term \(4x\) indicates a linear component within the quadratic expression.
• Quadratic functions are useful for modeling situations where there is acceleration, such as the path of a projectile.

Linear functions are the simplest form of polynomial functions, represented by an equation of the first degree, generally written as \(g(x) = mx + b\), where \(m\) and \(b\) are constants. These functions graph as straight lines, and their primary characteristic is a constant rate of change, which we see as the slope of the line.

The function \(g(x) = x + 2\) in our problem is linear. When a linear function is part of a composite function like in \(g \circ f\), it takes the output of another function (like our quadratic \(f(x)\)) and translates it, without affecting its degree.

• The slope \(m\) in our case is 1, indicating a 45-degree angle if plotted.
• The \(y\)-intercept \(b\) is 2, telling us where the line crosses the \(y\)-axis.
• Linear functions are crucial for understanding direct relationships between variables.