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Magazine Chapter 4: Problem 10
Determine whether the function \(f\) is one-to-one. $$f(x)=\sqrt[3]{x}$$
Short Answer
Expert verified
The function \(f(x) = \sqrt[3]{x}\) is one-to-one.
Step by step solution
01
Understand the Definition of One-to-One Function
A function is one-to-one if each input produces a unique output, meaning no two different inputs can have the same output. Formally, this means that if \(f(a) = f(b)\), then \(a = b\). We need to check this condition for the function \(f(x) = \sqrt[3]{x}\).
02
Set Up the Equation to Test One-to-One Property
Assume \(f(a) = f(b)\), which implies \(\sqrt[3]{a} = \sqrt[3]{b}\). Since the cube root function is the inverse of cubing, our goal is to show that \(a = b\) whenever their cube roots are equal.
03
Solve the Equation for a and b
If \(\sqrt[3]{a} = \sqrt[3]{b}\), then cubing both sides gives us \(a = b\). This follows from the cube root function's properties, which means that the function \(f(x) = \sqrt[3]{x}\) is indeed one-to-one because the input values \(a\) and \(b\) must be the same for their outputs to be equal.
04
Conclusion
Since \(a = b\) whenever \(\sqrt[3]{a} = \sqrt[3]{b}\), \(f(x) = \sqrt[3]{x}\) satisfies the condition for being a one-to-one function.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Function Properties
Function properties are foundational in understanding how functions behave and interact. A function, in mathematics, is a relation that maps every element from a set, called the domain, to exactly one element in another set, called the range.
Characteristics of functions include:
Characteristics of functions include:
- Domain: The set of all possible input values.
- Range: The set of all possible output values.
- Injective (One-to-One): A function is injective if different inputs always result in different outputs. No two different elements in the domain map to the same element in the range.
- Surjective (Onto): A function is surjective if every element of the range is mapped by at least one element of the domain.
Cube Root Function
The cube root function is a special type of function that can be expressed as \( f(x) = \sqrt[3]{x} \). It's a basic radical function and is essential in solving equations involving cubes.
Key aspects of the cube root function include:
Key aspects of the cube root function include:
- Domain and Range: Both the domain and range of the cube root function are all real numbers, meaning the function is continuous and can handle positive, negative, and zero values.
- Graph: The graph of the cube root function is symmetric about the origin. It passes through the origin (0,0), indicating that when x is 0, f(x) is also 0.
- Behavior: The cube root function increases steeply at the origin but levels out as x moves away from 0.
Inverse Functions
Inverse functions reverse the process of a given function. They 'undo' what the original function does by swapping the roles of inputs and outputs.
To determine if a function has an inverse, check if it is one-to-one. For instance, the cube root function \( f(x) = \sqrt[3]{x} \) is one-to-one, meaning it has an inverse, which is the cubing function \( g(x) = x^3 \). Other qualities of inverse functions include:
To determine if a function has an inverse, check if it is one-to-one. For instance, the cube root function \( f(x) = \sqrt[3]{x} \) is one-to-one, meaning it has an inverse, which is the cubing function \( g(x) = x^3 \). Other qualities of inverse functions include:
- Reflection: The graph of an inverse function is the reflection of the graph of the original function over the line \( y = x \).
- Algebraic confirmation: If \( y = f(x) \), then solving for \( x \) in terms of \( y \) gives the inverse function.
- Existence: Not all functions have inverses. They must be both injective and surjective.
Mathematical Proofs
Mathematical proofs provide a logical framework to validate statements and properties in mathematics. When proving that a function is one-to-one, like the cube root function \( f(x) = \sqrt[3]{x} \), proofs rely on logical reasoning supported by mathematical laws.
Steps involved in a typical proof include:
Steps involved in a typical proof include:
- Understand the assertion: Recognize that the proof aims to demonstrate a specific property, such as the injectivity of a function.
- Set up the assumption: Typically begin by assuming two separate elements yield the same image in the function.
- Manipulate mathematically: Use algebraic operations to show that these elements must be equal, substantiating the one-to-one property.
- Conclude: Clearly articulate the final position, which corroborates the initial assertion based on logical deduction.
