91Ó°ÊÓ

In mathematics, a quadratic equation is any polynomial equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \). The highest degree of the variable \( x \) is 2, which means these equations are represented by a parabola when graphed. In our case, the function \( f(x) = x^2 + x \) is a quadratic equation with specific constraints.To fully understand quadratic equations, it's important to recognize their structure and how they behave:

• The coefficient \( a \) defines the curve's direction; if positive, the parabola opens upwards, if negative, downwards.
• The value of \( b \) affects the symmetry and position of the vertex along the x-axis.
• \( c \) is the y-intercept, where the curve meets the y-axis.

In our given function, the quadratic nature can be observed from the \( ax^2 \) term, making solving for \( x \) possible using the quadratic formula. The domain restriction \( x \geq -\frac{1}{2} \) limits the part of the parabola we are interested in, which is essential when finding the inverse.

The domain and range of a function are fundamental concepts in mathematics, helping us understand which values belong to the input and output of a function.The domain of a function is the complete set of possible input values (often \( x \) values) for which the function is defined. For \( f(x) = x^2 + x \), the domain \( x \geq -\frac{1}{2} \) restricts us to consider only non-negative sections of \( x \) starting at \( -\frac{1}{2} \). This restriction ensures that the function remains injective (one-to-one), a necessary property for it to have an inverse.Conversely, the range is the set of possible output values (or \( y \) values). As the function \( x^2 + x \) is a parabola that opens upwards, its range starts from the minimum value it reaches when \( x = -\frac{1}{2} \). Calculating at this point, the range becomes \( y \geq -\frac{1}{4} \). These concepts are crucial when defining or finding an inverse function since they dictate the behavior of the inverse over its domain.

The quadratic formula is a powerful tool for solving quadratic equations. It's given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula solves any quadratic equation of the form \( ax^2 + bx + c = 0 \). By substituting the coefficients \( a \), \( b \), and \( c \) from the given equation, you obtain the values of \( x \) that satisfy the equation.In our example, we have:

• \( a = 1 \)
• \( b = 1 \)
• \( c = -y \)

Substitute these into the quadratic formula:\[x = \frac{-1 \pm \sqrt{1 - 4 \cdot 1 \cdot (-y)}}{2 \cdot 1} = \frac{-1 \pm \sqrt{1 + 4y}}{2}\]This gives us two solutions. For the inverse function, due to the domain's restriction \( x \geq -\frac{1}{2} \), we take the positive root \( x = \frac{-1 + \sqrt{1 + 4y}}{2} \).The quadratic formula is an essential tool in algebra, especially useful for finding roots of quadratic equations when factoring is impractical or impossible. Understanding this method is vital for solving equations and finding inverses.