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Chapter 5: Problem 48

Determine whether each function is one-to-one. If so, find its inverse. $$g=\left\\{\left(0,0^{4}\right),\left(-1,(-1)^{4}\right),\left(-2,(-2)^{4}\right),\left(-3,(-3)^{4}\right)\right\\}$$

Short Answer

Expert verified
The function is one-to-one, and its inverse is \( g^{-1} = \{(0,0), (1,-1), (16,-2), (81,-3)\} \).

Step by step solution

01

Understanding the Function

The function given is a set of ordered pairs: \( g=\{(0,0),(鈭�1,1),(鈭�2,16),(鈭�3,81)\} \). Each element of domain has a unique corresponding element in the codomain.
02

Check for One-to-One Property

A function is one-to-one if different inputs have different outputs. In \( g \), each input value (0, -1, -2, -3) corresponds to a unique output value (0, 1, 16, 81), so the function is indeed one-to-one.
03

Find the Inverse Function

For the inverse, interchange the elements in each pair: \( (0,0) \rightarrow (0,0), (1,-1) \rightarrow (1,-1), (16,-2) \rightarrow (16,-2), (81,-3) \rightarrow (81,-3) \). The inverse function is \( g^{-1} = \{(0,0), (1,-1), (16,-2), (81,-3)\} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inverse Functions
An inverse function essentially reverses the effects of the original function. Imagine having a machine that takes an input and produces an output. The inverse function acts like running the machine backward. For a function to have an inverse, it must be one-to-one, meaning each output is the result of exactly one input. If you can swap the input and output in all pairs and still have a function, then you've found the inverse. In the given exercise, our function is one-to-one, allowing us to find an inverse by flipping the ordered pairs.
Ordered Pairs
Ordered pairs are used to represent the input and output of a function. They are written as \((x, y)\), with \(x\) as the input and \(y\) as the output. In our exercise, the ordered pairs were \((0,0), (-1,1), (-2,16), (-3,81)\). Each pair shows how the function transforms the input into the output.
By interchanging the elements in these pairs, we can create the inverse function's pairs, turning \((0,0)\) into \((0,0)\), \((1,-1)\) into \((1,-1)\), and so on. This swap gives us a new set of ordered pairs that make up the inverse function.
Algebra
In the context of functions and their inverses, algebra helps us to understand and manipulate these relationships. Algebra provides the tools to express these ordered pairs and functions logically and systematically. It helps you understand the connections between numbers and variables, allowing for operations like adding, subtracting, or finding inverses.
When determining if a function has an inverse, algebraic methods help verify whether each input corresponds uniquely to one output. In this exercise, the calculation \((-1)^4 = 1\), \((-2)^4 = 16\), and so on, confirms the function's outputs as unique.
Function Properties
Certain properties determine whether a function is one-to-one. A function will be one-to-one if different inputs always produce different outputs. This property is crucial for finding an inverse. In our example, each input \(0, -1, -2, -3\) produces distinct outputs \(0, 1, 16, 81\), ensuring the function is one-to-one.
Identifying these properties comes down to checking the outputs for uniqueness. If ANY output is repeated for different inputs, the function can鈥檛 have an inverse. Understanding these properties not only helps in finding inverses but also in analyzing and simplifying functions in general.

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Most popular questions from this chapter

Using the restrictions on the functions in Exercises \(123-126,\) find a formula for \(f^{-1}\). $$f(x)=-x^{2}+4, \quad x \geq 0$$

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