Question

Find the domain of each function. \(f(x)=\sqrt[3]{5 x-4}\)

Short answer

The domain is \( (-\infty, \infty) \).

Step-by-step solution

- Understand the Function Type

Identify the type of function given. The function provided is a cube root function: \( f(x) = \sqrt[3]{5x - 4} \).

- Determine Restrictions

Cube root functions can accept any real number as input because the cube root of any real number is defined. Therefore, \( 5x - 4 \) can be any real number.

- Write the Domain

Since there are no restrictions on the input values of \(x\) for a cube root function, the domain is all real numbers.

- Express the Domain

Express the domain in interval notation. The domain of the function \( f(x) = \sqrt[3]{5 x - 4} \) is: \( (-\infty, \infty) \).

Key concepts

cube root function

A cube root function is a type of radical function that involves the cube root of a variable expression. In mathematical notation, a cube root function might look like this: \( f(x) = \sqrt[3]{expression} \).
The cube root of a number \( x \) is a value that, when multiplied by itself three times, gives \( x \). For example, the cube root of 8 is 2 because \( 2 \times 2 \times 2 = 8 \).
Cube root functions are fascinating because they are defined for all real numbers. Unlike square root functions, which are only defined for non-negative numbers (since you can't take the square root of a negative number in the set of real numbers), cube root functions don't have this restriction. This means the value inside the cube root can be negative, zero, or positive, and the function will still work.
For example, for the function \( f(x) = \sqrt[3]{5x - 4} \), you can substitute any real number for \( x \) and get a real number as a result. Because of this flexibility, cube root functions are quite versatile.

real numbers

Real numbers are one of the most fundamental sets of numbers in mathematics. They include every number that you can find on the number line. This means all positive numbers, negative numbers, and zero.
More formally, real numbers consist of:

• Rational numbers (those that can be expressed as a fraction of two integers, such as \( \frac{1}{2} \) or 0.75).
• Irrational numbers (those that cannot be expressed as a fraction of two integers, such as \( \sqrt{2} \) or \( \pi \)).

Real numbers are crucial when determining the domain of functions, especially for cube root functions. Since cube root functions can accept any real number as input, knowing what real numbers encompass helps in setting the correct domain.
For example, when we consider the domain of \( f(x) = \sqrt[3]{5x - 4} \), we don't need to worry about exclusions or restrictions because every component in that function will work with any real number substituted for \( x \).

interval notation

Interval notation is a way of writing subsets of the real number line. It is efficient and clear, especially when expressing domains and ranges of functions.
There are a few key symbols used in interval notation:

• Parentheses \( () \) indicate that the endpoints are not included in the interval.
• Brackets \( [] \) indicate that the endpoints are included in the interval.
• Infinity (\( +\infty \)) and negative infinity (\( -\infty \)) are used to express unbounded sections of the number line.

For example, the interval \( (a, b) \) includes all numbers between \( a \) and \( b \) but not including \( a \) and \( b \). On the other hand, \( [a, b] \) includes both \( a \) and \( b \) along with all the numbers in between.
In the case of our function \( f(x) = \sqrt[3]{5x - 4} \), since the cube root function can accept any real number as input, the domain is the entire set of real numbers. We express this as \( (-\infty, \infty) \). This notation succinctly says that \( x \) can be any real number from negative infinity to positive infinity, reflecting the unrestricted input values for a cube root function.