91Ó°ÊÓ

A quadratic equation is an equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). This type of equation is characterized by the highest exponent of the variable, usually \( x \), being 2. Quadratics are pivotal in determining the roots or solutions where the function intersects the x-axis.

To solve a quadratic, we can use various methods:

• The quadratic formula, given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• Factoring, if the quadratic can be expressed as a product of linear factors.
• Completing the square, which involves transforming a quadratic into a perfect square trinomial.

This equation format allowed us to express the inverse function’s formula after rearranging the original function. By recognizing the quadratic form of \( 5x^2 - x + y = 0 \), we realized it was solvable using the quadratic formula, which provided the bridge to isolating \( x \) for the inverse.

The domain of a function is the complete set of possible values of the independent variable (often \( x \)) for which the function is defined. It's important as it determines the range over which the function can operate without generating undefined or nonsensical outputs. For a function's inverse, the domain plays a crucial role because it signifies the values for which the inverse remains valid.

When we find the inverse of a function, its domain is derived based on the original function's range. In our case, to ensure the expression under the square root, \( 1 - 20x \), remains non-negative, we need \( x \leq \frac{1}{20} \). Thus, the domain of \( f^{-1}(x) \) is \( x \in [0, \frac{1}{20}] \), ensuring all values work in the inverse's formula.

The square root is a mathematical operation that finds a number which, when multiplied by itself, results in the original number. For example, the square root of 16 is 4, because \( 4 \times 4 = 16 \). A vital property of the square root operation is that it returns non-negative results when applied to non-negative inputs.
The function \( \sqrt{x} \) is defined only when \( x \geq 0 \), making it significant in determining the domain of functions involving square roots.

In dealing with inverse functions, ensuring the expression inside the square root is non-negative is key. In our exercise, \( \sqrt{1-20x} \) needed to exist only for \( 1-20x \geq 0 \). Respecting this constraint allowed us to find the correct domain of the inverse function, ensuring all mathematical operations within it were valid and real.