91Ó°ÊÓ

When we refer to composite functions, we are talking about functions composed of other functions, essentially 'functions within functions'. The notation \( (fg)(x) \) represents the composite function which is the result of applying \( g \) to \( x \) first and then applying \( f \) to the result. To find the domain of the composite function, we need to ensure that every \( x \) not only lies in the domain of \( g \) but also that \( g(x) \) lies in the domain of \( f \).
In our exercise, because both \( f \) and \( g \) functions are defined for all real numbers, their composite function \( f(g(x)) \) will also be well-defined for all real numbers (\(\mathbb{R}\)). Thus, the domain of the composite function \( fg \) is \( x \in \mathbb{R} \).

The 'quotient of functions' is expressed by dividing one function by another, written as \( \frac{f(x)}{g(x)} \) or \( f / g \). To determine the domain of this quotient, we need to consider where the denominator, \( g(x) \) is nonzero, because division by zero is undefined. Essentially, the domain of \( f / g \) will be all the values in the domain of \( f \) that do not make \( g(x) \) equal zero.
In our specific case, \( g(x) = 4x - 3 \) cannot equal zero, so we solve for when \( 4x - 3 eq 0 \) to find where the quotient \( f/g \) is defined. This gives us all real numbers except \( x = \frac{3}{4} \) as the domain of the quotient \( f / g \), which can be stated as \( \mathbb{R} \setminus \{\frac{3}{4}\} \) or \( x \in \mathbb{R}, x eq \frac{3}{4} \).

Polynomial functions, like the one in \( f(x) = 3x^2 + x^4 + 2 \) from our exercise, are algebraic expressions that consist of terms involving variables raised to whole number exponents summed together. Polynomial functions are continuous and smooth, which means they do not have breaks, jumps, or sharp corners. The domain of a polynomial function is the set of all real numbers (\(\mathbb{R}\)) because you can plug any real number into a polynomial and get a real number out.
Therefore, the domain of \( f(x) \) in our problem is all real numbers, which shows its extensive range of possible inputs.

Linear functions are the simplest form of polynomial functions, typically represented by \( g(x) = ax + b \) with \( a \) and \( b \) as constants. These functions graph as straight lines and they have a constant rate of change, which is the slope. The domain of linear functions is also all real numbers (\(\mathbb{R}\)), as there are no restrictions on the values that \( x \) can take. This characteristic is displayed in the \( g(x) \) function from our original task, \( g(x) = 4x - 3 \), which has no restrictions on its input. Consequently, the domain is the entirety of real numbers, reinforcing the inclusivity of linear functions' domains.