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Quadratic functions are equations that can be expressed in the form \( y = ax^2 + bx + c \). They are characterized by their U-shaped graphs, known as parabolas. These functions may open upward or downward based on the sign of the leading coefficient \( a \). If \( a > 0\), the parabola opens upward, whereas \( a < 0 \) opens it downward. In this exercise, the function \( y = x^2 + 2 \) is a simple quadratic with \( a = 1 \), so it opens upwards.

There are several key features of quadratic functions worth noting:

• The vertex is the highest or lowest point on the graph. In our function, the vertex is at the point where \( x = 0 \), yielding \( y = 2 \).
• The axis of symmetry, a vertical line through the vertex, divides the parabola into two mirror-image halves. For the function given, this line is \( x = 0 \).

The one-to-one property of a function ensures that each input has a unique output, and vice versa. For a function to have an inverse, it must be one-to-one. Quadratic functions are not inherently one-to-one because they typically cover values in both positive and negative directions.

However, by restricting the domain, we can make a quadratic function one-to-one. In this case, the domain is limited to \( 0 \leq x \leq 5 \), making the function \( y = x^2 + 2 \) one-to-one over this interval. Here, the function is strictly increasing because we only consider \( x \geq 0 \). This allows each \( y \) value to have a unique corresponding \( x \) value, enabling the possibility of an inverse.

A bijective function is one that is both injective (one-to-one) and surjective (onto). When a function satisfies both these conditions, it means every \( y \) value in the range has exactly one corresponding \( x \) value in the domain.

In this exercise, the function \( y = x^2 + 2 \) defined for \( 0 \leq x \leq 5 \) is injective because of the restricted domain. Each \( x \) maps to a unique \( y \), satisfying the injective condition. The function is also surjective onto its range \( [2, 27] \), meaning that for every \( y \) in this range, there exists an \( x \) such that \( y = x^2 + 2 \). Hence, this function is bijective over the given domain and range.

The domain of a function is the set of all possible input values \( x \), while the range is the set of all possible output values \( y \). Identifying these is crucial for understanding how a function behaves.

For the function \( y = x^2 + 2 \) with \( 0 \leq x \leq 5 \), the domain is straightforwardly \([0, 5]\), which means that the input \( x \) can take any value from 0 to 5, inclusive. The range is determined by the behavior of the quadratic expression. With \( y = x^2 + 2 \), when \( x = 0 \), \( y = 2 \), and when \( x = 5 \), \( y = 27 \). Therefore, the range is \([2, 27]\), encompassing all possible output values between these limits. This understanding is crucial when determining if a function has an inverse, as it helps to ensure the inverse function properly maps from range back to domain.