Problem 53 - Algebra A Combined Function

Question

Solve. See the Concept Check in this section. Suppose that \(f\) is a one-to-one function and that \(f(2)=9\) a. Write the corresponding ordered pair. b. Name one ordered pair that we know is a solution of the inverse of \(f\), or \(f^{-1}\).

Step-by-Step Solution

Answer

a. (2, 9); b. (9, 2)

Step 1: Understand the Function

A function is a mapping from elements of one set to elements of another set. In this problem, the function \(f\) is one-to-one, meaning each output is uniquely paired with one input.

Step 2: Write the Ordered Pair for the Function

We are given that \(f(2) = 9\). This means when the input is 2, the output is 9. We express this relationship as an ordered pair: \((2, 9)\).

Step 3: Understand the Inverse Function

The inverse function, \(f^{-1}\), reverses the roles of inputs and outputs. If \(f(a) = b\), then \(f^{-1}(b) = a\). This is true because the inverse function undoes the work of the original function.

Step 4: Determine the Ordered Pair for the Inverse

Using the relationship \(f(2) = 9\), we can determine a corresponding pair for the inverse function. Because \(f(2) = 9\), then \(f^{-1}(9) = 2\). This inverse pair is expressed as \((9, 2)\).

Key Concepts

One-to-One Functions

A one-to-one function is a special type of function in mathematics where each element of the function's domain (input set) corresponds to a unique element in its codomain (output set). This unique correspondence means that no two different inputs map to the same output. In simpler terms, a one-to-one function never assigns the same value in the range to two different domain elements.

• For example, if we know that a function maps 2 to 9, such that \(f(2) = 9\), we can confidently say that there isn't another input, say 3, that can also be mapped to 9.
• One-to-one functions are often denoted as injective functions in formal mathematical language.

When dealing with inverses, recognizing a function as one-to-one confirms that the inverse is also well-defined, ensuring that every output from the function is original and can be traced back uniquely to one input from the domain.

Ordered Pairs

Ordered pairs are a fundamental concept in mathematics used to describe relationships between two elements. An ordered pair is written in the form \((a, b)\), where "\(a\)" is generally the input or independent variable, and "\(b\)" is the output or dependent variable.

• The order of elements in an ordered pair is critical because it defines the relationship's direction.
• For functions, ordered pairs illustrate how each input maps to a unique output, such that if \(f(x) = y\), the ordered pair will be \((x, y)\).

In the current context, the ordered pair \((2, 9)\) depicts that when the input is 2, the output of the function is 9. The clear depiction of this relationship helps understand how functions operate and how they can be reversed through inverse functions.

Inverse Relationship

An inverse relationship in mathematics involves swapping the roles of inputs and outputs in a function. When a function \(f\) is one-to-one, it has an inverse, denoted as \(f^{-1}\). This inverse essentially "undoes" the action of the original function.

• For a function \(f\) that maps \(x\) to \(y\) (\(f(x) = y\)), its inverse \(f^{-1}\) will map \(y\) back to \(x\) (\(f^{-1}(y) = x\)).
• This swapping showcases a two-directional relationship where the original and inverse functions complement each other.

In practice, understanding the inverse relationship means recognizing an ordered pair like \((2, 9)\) will lead to the inverse pair \((9, 2)\). This reverse mapping is crucial in mathematical problems involving inverse functions.

Function Mapping

Function mapping is the process by which elements of one set (the domain) are paired with elements in another set (the codomain) through a function. Each element from the domain is mapped to exactly one element in the codomain, following the definition of a function.

• In the context of one-to-one functions, function mapping is unique and precise, meaning each input corresponds to one output only.
• This mapping is visually represented through ordered pairs \((x, y)\), indicating that \(x\) maps to \(y\).

When dealing with inverse functions, this mapping is reversed. The inverse mapping shows the relationship going from the output back to the input, fully realizing the function's capability to transition between these two sets. In our example, the pair \((2, 9)\) is mapped directly by \(f\), while its inverse mapping \((9, 2)\) is accomplished by \(f^{-1}\). Understanding function mapping is vital for grasping how functions and their inverses operate within mathematical systems.