For let f(x)=(a x+b)/(c x+d). (a) Show that f is one-to-one...

Key Concepts

Finding the Inverse Function

Once it is established that a function is one-to-one (injective), the next step is to find its inverse function. This involves solving the equation y = (ax+b)/(cx+d) for x in terms of y. The process typically includes cross-multiplying and isolating x, and it is important to impose the condition found earlier (ad - bc ? 0) to ensure that each algebraic step is valid and the inverse function exists. This technique is widely used in algebra and calculus to derive the inverse of a given function.

Injective Functions

An injective function, often referred to as one-to-one, is a function where each element of the codomain is mapped by at most one element of the domain. This concept ensures that no two different inputs produce the same output, which is a critical property for a function to have an inverse. In proving injectivity, one typically sets f(x1) equal to f(x2) and shows that this leads to x1 = x2 under the given conditions.

Rational Functions

Rational functions are ratios of polynomial functions, and their behavior can be quite complex depending on the degrees and coefficients of the polynomials. In this context, the function f(x) = (ax+b)/(cx+d) is a common example that often shows up as a Möbius transformation in advanced mathematics. Studying its properties, such as domain restrictions, asymptotic behavior, and invertibility, is fundamental in understanding more advanced function theory.

Determinant Condition for Invertibility

The condition ad - bc ? 0 is crucial in ensuring that the function f(x) = (ax+b)/(cx+d) is invertible. This condition is analogous to the determinant of a 2x2 matrix being non-zero, which implies that the matrix (or transformation) is nonsingular and hence invertible. In this context, the non-zero determinant guarantees that the function is one-to-one, making it possible to define a unique inverse function.

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