What is the domain and range of(2x^5-1)^1/3

To determine the **domain** and **range** of the function \( f(x) = \left(2x^5 - 1\right)^{\frac{1}{3}} \), let's break it down step by step:

### Domain
The domain of a function refers to all the values of \( x \) for which the function is defined.

For the function \( f(x) = \left(2x^5 - 1\right)^{\frac{1}{3}} \), note that:
- The expression \( 2x^5 - 1 \) can take any real value, because polynomials (like \( 2x^5 - 1 \)) are defined for all real \( x \).
- The cube root function \( (\cdot)^{\frac{1}{3}} \) is defined for all real numbers, including both positive and negative values.

Thus, there are **no restrictions** on \( x \). Therefore, the domain of \( f(x) \) is:

\[
\boxed{\mathbb{R}} \quad \text{(All real numbers)}
\]

### Range
The range of a function refers to all the possible output values (i.e., values of \( f(x) \)).

- The expression \( 2x^5 - 1 \) can take any real value, as the term \( 2x^5 \) grows without bound as \( x \) increases or decreases.
- The cube root of any real number is also a real number, which means that \( \left(2x^5 - 1\right)^{\frac{1}{3}} \) can also take any real value.

Thus, the range of \( f(x) \) is also:

\[
\boxed{\mathbb{R}} \quad \text{(All real numbers)}
\]

### Summary
- **Domain:** \( \mathbb{R} \) (all real numbers)
- **Range:** \( \mathbb{R} \) (all real numbers)

Would you like further details or have any questions?

Here are 5 related questions:
1. What is the domain and range of \( (3x^4 + 2x - 5)^{1/2} \)?
2. How do you find the range of a function given its graph?
3. What is the inverse of \( (2x^5 - 1)^{1/3} \)?
4. What happens to the domain when dealing with even roots?
5. How do you solve equations involving cube roots?

**Tip:** Cube root functions are defined for all real numbers, unlike square roots which are limited to non-negative numbers.

What is the domain and range of (2x^5 - 1)^(1/3)?

Explore how to determine the domain and range of the function (2x^5 - 1)^(1/3) using algebraic concepts. Learn the key properties of polynomials and cube roots, with step-by-step guidance. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation