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A constant function is one of the simplest types of functions you'll encounter in mathematics. For a given input, it always provides the same output value, which means it is a function that doesn't change. In our example, we have the function \( f(x) = 2 \), where no matter what value of \( x \) you plug into the function, the output will always be 2.

The domain of a constant function, such as \( f(x) = 2 \), is all real numbers, \( \mathbb{R} \). Why? Because there is no restriction or undefined point at any value of \( x \). This means you can plug in any number, whether positive, negative, or fractional, and \( f(x) \) will still be valid.

The range of a constant function, on the other hand, is quite specific. Since the output of \( f(x) = 2 \) is always 2, the range is just the single value \( \{2\} \). Understanding this property of constant functions helps in simplifying and analyzing function behaviors, especially when they are part of more complex function operations.

Quadratic functions are a bit more complex than constant functions but still follow certain patterns. A standard quadratic function can be expressed as \( g(x) = ax^2 + bx + c \). It forms a parabola when plotted on a graph.

In our exercise, the quadratic function is \( g(x) = x^2 + 1 \). This means the parabola opens upwards, and it is shifted one unit above the x-axis. The domain of \( g(x) = x^2 + 1 \) includes all real numbers, \( \mathbb{R} \), because no matter what real number you choose for \( x \), you can calculate \( g(x) \) without running into issues like division by zero.

Next is the range. The function \( g(x) = x^2 + 1 \) achieves its minimum value when \( x = 0 \). This leads to the minimum output of 1. The values of \( g(x) \) can get as large as you want as \( x \) increases positively or negatively, due to the \( x^2 \) term. Therefore, the range is \( [1, \infty) \), meaning the function covers all values from 1 upwards.

Function division involves dividing one function by another, resulting in a new function. Consider the functions \( f(x) = 2 \) and \( g(x) = x^2 + 1 \). When dividing \( f(x) \) by \( g(x) \), we get \( \frac{f(x)}{g(x)} = \frac{2}{x^2 + 1} \).

It's important to determine the domain of this division. For \( \frac{f(x)}{g(x)} \), check if there's any \( x \) at which the denominator would be zero, making the function undefined. But since \( x^2 + 1 = 0 \) has no real solutions, \( x^2 + 1 \) is never zero; thus, the domain is \( \mathbb{R} \).

The range of this function division is interesting. The largest value is \( 2 \) when \( x = 0 \). As \( x^2 \) increases, \( x^2 + 1 \) also increases, causing \( \frac{2}{x^2 + 1} \) to decrease. As a result, the range shrinks from the value \( 2 \) downward, approaching zero but never actually reaching it. This gives the range \( (0, 2] \).

Another division to consider is \( \frac{g(x)}{f(x)} = \frac{x^2 + 1}{2} \), where \( f(x) = 2 \) is a constant that simplifies the division. This function has the domain \( \mathbb{R} \) again, and the range starts from \( 0.5 \) upward because the minimum value achieved is \( \frac{1}{2} \). The expression grows without bounds, leading to a range of \( [0.5, \infty) \).