91Ó°ÊÓ

Quadratic functions are a type of polynomial function and are generally represented in the form \(f(x) = ax^2 + bx + c\). In these equations:

• \(a\), \(b\), and \(c\) are constants.
• The coefficient \(a\) must be non-zero (\(a eq 0\)); otherwise, the function would no longer be quadratic.
• The quadratic term \(ax^2\) gives the function its characteristic parabolic shape.

The shape of the parabola depends on the sign of the leading coefficient \(a\). If \(a > 0\), the parabola opens upwards, and if \(a < 0\), it opens downwards. The vertex of the parabola, which is the point of minimum or maximum value, can be found using the formula \(x = -\frac{b}{2a}\). This vertex is essential in calculus for analyzing the behavior of the function.Additionally, quadratic functions have distinctive properties such as symmetry around the axis parallel to the \(y\)-axis that passes through the vertex, and they can model a variety of real-world phenomena including projectile motion and area problems. Understanding how to evaluate values of a quadratic function at specific points, like [1, 2) in our exercise, helps determine the range of composed functions.

The fractional part function, often represented as \(\{x\}\), extracts the decimal portion of a number. This function is defined as:

• \(\{x\} = x - \lfloor x \rfloor\) where \(\lfloor x \rfloor\) is the greatest integer less than or equal to \(x\).

In simpler terms, for any given number, the fractional part function will take away the whole number part (or integer part), leaving only the fraction or decimal.For instance, with \(x = 1.75\), we have:

• Whole part: \(\lfloor 1.75 \rfloor = 1\)
• Fractional part: \(\{1.75\} = 1.75 - 1 = 0.75\)

The fractional part is always in the range \([0, 1)\), meaning it can be zero but not one. In our exercise, when applying the function \(g(x) = \{x\} + 1\), it leads to values strictly between 1 and 2, clarifying the domain of \(f(g(x))\). Understanding this helps simplify expressions and calculate more complex functions involving repetition or periodicity of decimal values.

Function composition involves creating a new function by applying one function to the results of another, denoted as \((f \circ g)(x) = f(g(x))\). This is an important concept in calculus and helps solve complex equations by breaking them into manageable parts.

• The function \(g(x)\) is applied first to the input \(x\).
• The result of \(g(x)\) is then used as the input for the function \(f(x)\).

In our exercise, \(f(x) = x^2 - x + 2\), a quadratic function, is composed with \(g(x) = \{x\} + 1\), a function based on the fractional part. The composition \(f(g(x))\) evaluates the transformation of the domain \([1, 2)\) through both functions.An important aspect of function composition is that the range of \(g(x)\) becomes the domain of \(f(x)\), which helps define both the sequence of transformations and resulting outputs. Understanding the process of substitution and transformation in each step of composing functions not only simplifies complex problems but also deepens comprehension of how individual elements interact within a function's framework.