Problem 54 - Algebra A Combined Function

Question

Solve. See the Concept Check in this section. Suppose that \(F\) is a one-to-one function and that \(F\left(\frac{1}{2}\right)=-0.7\) a. Write the corresponding ordered pair. b. Name one ordered pair that we know is a solutic of the inverse of \(F,\) or \(F^{-1}\).

Step-by-Step Solution

Answer

(\frac{1}{2}, -0.7) and (-0.7, \frac{1}{2}) are the ordered pairs for F and F^{-1}, respectively.

Step 1: Identify Ordered Pair

Since the function \(F\) is given as \(F\left(\frac{1}{2}\right)=-0.7\), we can identify the ordered pair corresponding to this information. In the context of function notation, the input \(x\) maps to the output \(y\) as \((x, y)\). Thus, the ordered pair for the function \(F\) is \(\left(\frac{1}{2}, -0.7\right)\).

Step 2: Determine Ordered Pair for Inverse Function

By the definition of inverse functions, if \(F(a) = b\), then \(F^{-1}(b) = a\). Thus, the ordered pair \((b, a)\) corresponds to the inverse function \(F^{-1}\). Given \(F\left(\frac{1}{2}\right) = -0.7\), the ordered pair for \(F^{-1}\) is \((-0.7, \frac{1}{2})\).

Key Concepts

One-to-One Functions

A one-to-one function, also known as an injective function, has a special property where each input value is mapped to one distinct output value. This means no two different input values will produce the same output. For example, if a function says \( F(a) = b \) and \( F(c) = b \), then it must be true that \( a = c \).
This unique mapping makes one-to-one functions quite useful because they guarantee that each element of the function's range is paired with exactly one element of its domain.

• Ensures that for each output, there is a unique input.
• Critical for determining the inverse of a function.

Understanding this property helps in identifying relationships where inverse functions can easily be found.

Ordered Pairs

Ordered pairs represent the relationship between two elements in a function's domain and range. It's like having a coordinate on a graph where the first value is the x-coordinate (input) and the second is the y-coordinate (output). In function notation, an ordered pair \(x, y\) indicates that when x is input into the function, y is the resulting output.
In our example, the ordered pair \( \frac{1}{2}, -0.7 \) was derived because when \( \frac{1}{2} \) is input into the function \( F \), the output is \( -0.7 \).
For inverse functions, the roles of x and y switch, forming a new ordered pair. Hence, \(-0.7, \frac{1}{2} \) becomes an ordered pair in the inverse function \( F^{-1} \).

• Ordered pairs help visualize and understand functions and their inverses.
• They showcase the direct relationship between an input and output.

Function Notation

Function notation is a systematic way of showing the relationship between inputs and outputs using symbols and expressions. In essence, it provides a compact and clear way to denote functions.
For example, \( F(x) \) signifies a function named \( F \) with \( x \) as the variable. Here, \( F \) is like a machine that transforms the input \( x \) into an output based on a specific rule.
Understanding function notation is crucial as it simplifies the representation and manipulation of functions. It helps us quickly identify the relationships between variables and effectively express the inverse too.

• It provides a clearer alternative to traditional algebraic expressions.
• Aids in conveying complex dependencies in a straightforward manner.

Function notation is integral to learning about functions and their properties, like one-to-one mappings and inverse functions.