Problem 9 - Elementary and Intermediate Algebra

Question

Fill in the blanks. If \(f\) is a one-to-one function, and if \(f(1)=6,\) then \(f^{-1}(6)=\) ____

Step-by-Step Solution

Answer

The answer is 1.

Step 1: Understanding a One-to-One Function

A one-to-one function, also called an injective function, is a type of function where each element of the range is paired with exactly one element of the domain. In this scenario, it means that every output has a unique input.

Step 2: Interpreting Given Information

We are given that if a function \( f \) is one-to-one and \( f(1) = 6 \), then this tells us that when the input is 1, the output is 6.

Step 3: Understanding the Inverse Function

The inverse function \( f^{-1} \) of a one-to-one function \( f \) reverses the role of inputs and outputs. Basically, it takes outputs back to their corresponding inputs.

Step 4: Finding the Inverse Output

Given \( f(1) = 6 \), the inverse function \( f^{-1}(6) \) returns the original input that corresponds to the output 6. Since \( f \) maps 1 to 6, \( f^{-1} \) must map 6 back to 1.

Key Concepts

One-to-One Function

A one-to-one function is a special type of function that has a unique relationship between its inputs and outputs. In mathematical terms, this means that every "output" value in the function corresponds to one and only one "input" value. No two different inputs can produce the same output. This characteristic is crucial when it comes to defining an inverse function.

• If the function is not one-to-one, it cannot have an inverse that is also a function.
• Graphically, a one-to-one function passes the "horizontal line test" – no horizontal line should intersect the graph of the function more than once.
• The notation for a one-to-one function is often expressed as maintaining distinctness: if \(a eq b\), then \(f(a) eq f(b)\).

Recognizing a one-to-one function allows us to safely determine that each output has a specific input, which is foundational for the function's inverse.

Function Inversion

Function inversion is the process of swapping roles between inputs and outputs of a function. For a function to be inverted, it must be one-to-one. When you perform function inversion, the inverse function, denoted \(f^{-1}\), can be used to retrieve the original input from a given output.

• The process involves solving the equation \(y=f(x)\) for \(x\) in terms of \(y\), thus essentially "reversing" the function.
• An important property of inverses is that applying a function then its inverse returns the original value: \(f^{-1}(f(x)) = x\) and vice versa.
• Not every function has an inverse since only bijective (both injective and surjective) functions meet the criteria.

Understanding function inversion is vital as it provides a method to "undo" the effect of a function, retrieving original values based on what was mapped to them originally.

Injective Function

Injective functions, also widely known as one-to-one functions, are fundamental in understanding how function inversion works. When discussing injections, the focus is on the uniqueness of mapping from domain to range.

• Injective functions ensure that every element of the domain maps to a distinct element in the codomain.
• This property prevents two different elements of the domain from mapping to the same element in the range, maintaining a unique relation throughout the set.
• A mathematical expression for injectivity is: for every \(a\) and \(b\) in domain, if \(f(a) = f(b)\), then \(a = b\).

Grasping the concept of injective functions is significant because it guarantees that the inverse function will also be a valid and unique function, constructively contributing to the full understanding of function behavior in mathematical contexts.