Problem 70 - Algebra 2

Question

For what positive integers \(n\) are the domain and range of \(y=\sqrt[n]{x}\) the set of real numbers? Assume that \(x\) is a real number.

Step-by-Step Solution

Answer

The function \(y=\sqrt[n]{x}\) has both domain and range as the set of all real numbers if and only if \(n\) is a positive odd integer.

Step 1: Understanding the function

We start by understanding the function \(y=\sqrt[n]{x}\) or equivalently \(y = x^{1/n}\). The function 's domain is the set of all \(x\) such that \(y\) is a real number. By definition, \(y\) is the principal \(n\)-th root of \(x\). The principal \(n\)-th root of \(x\) is a real number if \(x\) is non-negative when \(n\) is even and is a real number if \(x\) is any real number when \(n\) is odd.

Step 2: Analyzing the even case

If \(n\) is an even integer, \(x\) must be non-negative for \(y\) to be a real number. This means if \(n\) is even, then the domain of the function is the set of all non-negative real numbers. Hence, for even \(n\) the function does not take all real numbers as input, thus they do not fulfill the requirements from the problem.

Step 3: Analyzing the odd case

If \(n\) is an odd integer, \(x\) can be any real number for \(y\) to be a real number. This means if \(n\) is odd, then the domain of the function is the set of all real numbers. Hence, for odd \(n\), the function can take any real number as input and has the range as all real numbers. Therefore, any positive odd integer \(n\) qualifies as a solution.

Key Concepts

Domain of Functions

The domain of a function defines all the possible inputs that the function can accept. When you think of a function like a machine, the domain is all the different types of materials you can feed into this machine.

Let's talk about the function in question, \( y = \sqrt[n]{x} \), which can be rewritten as \( y = x^{1/n} \). Here, you are dealing with the principal \( n \)-th root of \( x \). The function's domain changes based on whether \( n \) is an odd or even integer.

• **When \( n \) is even**: The domain is restricted to non-negative real numbers. This is because taking the even root (like a square root) of a negative number does not yield a real number.
• **When \( n \) is odd**: The domain opens up to all real numbers. Odd roots (like cube roots) can handle negative numbers just fine.

Understanding the distinction between even and odd functions is key to determining the domain.

Range of Functions

The term "range" of a function refers to all the possible outputs that the function can produce. It's like asking, "After putting something into the machine, what can come out?"

For the function \( y = \sqrt[n]{x} \), the range differs depending on the parity (odd or even) of \( n \).

• **Even \( n \)**: The range is limited to non-negative real numbers. When you take an even root of a non-negative number, you cannot get a negative result.
• **Odd \( n \)**: The range encompasses all real numbers. With odd roots, you can achieve both negative and positive values, meaning the full spectrum of real numbers is possible.

Thus, only when \( n \) is odd can the range be the entire set of real numbers.

Odd and Even Integers

Understanding integers as odd or even is crucial for functions involving \( n \)-th roots. Odd and even integers behave differently when used as exponents in root functions.

• **Even Integers**: Even integers are whole numbers divisible by 2, such as 2, 4, 6, and so on. They affect the domain and range of \( n \)-th root functions by limiting them to non-negative values.
• **Odd Integers**: Odd integers can handle negative numbers and they are numbers like 1, 3, 5, etc. When used as roots, they allow both the domain and range to span all real numbers.

When solving problems involving functions with roots, knowing whether \( n \) is odd or even is essential to understanding what values \( x \) can take and what values \( y \) can output.