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Function holes, sometimes referred to as removable discontinuities, occur at a specific point on the graph of a function where the function is not defined. This means there's an interruption or a 'hole' at that particular spot, although the rest of the function behaves normally. Consider a continuous line representing a function, but with a tiny gap at one point—this gap is the hole.

• Formation: Holes usually arise from a ratio of polynomials when the numerator and denominator both have a common factor that cancels out; however, in our exercise, the hole is intentionally created by designating a specific point as undefined.
• Identification: You can identify a hole by finding a value of \(x\) where the function should be undefined while elsewhere it maintains its original value.

In the provided problem, a hole at \(x = 0\) is created in the function \(g(x)\) to make it undefined at that point while matching \(f(x)\) otherwise. This hole is represented by the coordinates \((0, 1)\).

A continuous function is a function with no breaks, jumps, or holes throughout its domain. You can visualize this as a curve that you can draw on a graph without lifting your pencil.

• Definition: Mathematically, a function \(f\) is continuous at a point \(x = c\) if \(\lim_{{x \to c}}f(x) = f(c)\). It must be defined at \(c\).
• Graph: The graph of a continuous function is smooth and unbroken over its entire range where it is defined. For quadratic functions like \(f(x) = x^2 + 1\), it takes the shape of a parabola.

In our exercise, the function \(f(x) = x^2 + 1\) is a continuous function naturally, since it is defined for all real numbers. The absence of initial function holes signifies its continuity. However, deliberately making \(g(x)\) undefined at \(x = 0\) introduces a discontinuity at that single point.

Quadratic functions are polynomial functions of degree two, meaning the highest power of the variable is two. They take the general form \(f(x) = ax^2 + bx + c\) where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The graph of a quadratic function is a parabola, typically opening upwards if \(a > 0\) and downwards if \(a < 0\).

• Vertex: The vertex of the parabola is its highest or lowest point, depending on its direction. This can be found using the formula \(x = -\frac{b}{2a}\).
• Symmetry: Quadratic functions are symmetric about a vertical line that passes through the vertex. This line is called the axis of symmetry.

In our specific example, the function \(f(x) = x^2 + 1\) is a basic quadratic function without linear or higher degree terms. It simply produces a parabola with vertex at \((0, 1)\), and since \(a = 1\), it opens upwards. The function is continuous, and its simplicity helps to clearly illustrate the concept of introducing a function hole at \(x = 0\) in \(g(x)\).