Math Problem Statement

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

Solution

To determine the domain and range of the function f(x)=(2x51)13f(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 xx for which the function is defined.

For the function f(x)=(2x51)13f(x) = \left(2x^5 - 1\right)^{\frac{1}{3}}, note that:

  • The expression 2x512x^5 - 1 can take any real value, because polynomials (like 2x512x^5 - 1) are defined for all real xx.
  • The cube root function ()13(\cdot)^{\frac{1}{3}} is defined for all real numbers, including both positive and negative values.

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

R(All real numbers)\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)f(x)).

  • The expression 2x512x^5 - 1 can take any real value, as the term 2x52x^5 grows without bound as xx increases or decreases.
  • The cube root of any real number is also a real number, which means that (2x51)13\left(2x^5 - 1\right)^{\frac{1}{3}} can also take any real value.

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

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

Summary

  • Domain: R\mathbb{R} (all real numbers)
  • Range: R\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 (3x4+2x5)1/2(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 (2x51)1/3(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.

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Math Problem Analysis

Mathematical Concepts

Algebra
Functions
Domain and Range
Cube Roots

Formulas

f(x) = (2x^5 - 1)^(1/3)

Theorems

Domain of Polynomials
Properties of Cube Roots

Suitable Grade Level

Grades 10-12