Question

\(f(x)=\sqrt[3]{x}\)

Short answer

The function \(f(x)=\sqrt[3]{x}\) is a cube root function. Its graph is a curve that starts from the negative y-axis and increases to the positive x and y-axis. The domain and range of the function are all real numbers.

Step-by-step solution

Understanding Cube Roots

A cube root of a number \(x\) is a number which when cubed, gives \(x\). For example, the cube root of 8 is 2 because \(2^3=8\). In this case, the function \(f(x)=\sqrt[3]{x}\) means that for each \(x\), \(f(x)\) is the cube root of \(x\).

Graphing the Cube Root Function

The graph of the function \(f(x)=\sqrt[3]{x}\) is a curve that starts from the negative side of the y-axis and gradually rises towards the positive side of both the x-and y-axis. This graph is symmetric with respect to the origin (0,0).

Finding the Domain

The domain of a function is the set of all possible \(x\) values. For the cube root function, \(f(x)=\sqrt[3]{x}\), x can be any real number. Therefore, the domain is all real numbers or in interval notation, \((-\infty, \infty)\).

Finding the Range

The range of a function consists of all possible outputs or \(y\) values. For \(f(x)=\sqrt[3]{x}\), the range is also all real numbers or in interval notation, \((-\infty, \infty)\), because there is no limit to the values that \(f(x)\) can take.

Key concepts

Graphing Functions

Graphing functions, such as the cube root function\(f(x) = \sqrt[3]{x}\), helps us visualize how changes in \(x\) affect \(f(x)\). To plot the graph of a cube root function, imagine drawing a curve that snakes its way through the coordinate system. This particular curve starts below the x-axis where \(x\) is negative. This is because the cube root of a negative number is also negative. As \(x\) transitions from negative to positive values, the graph smoothly rises to the origin and then continues into the first quadrant of the coordinate plane.
The shape of this graph is interesting because it exhibits symmetry with respect to the origin, meaning if you rotate the graph 180 degrees around the origin, it looks exactly the same.
If you would like to try graph it yourself, just remember:

• The graph passes through the origin (0,0).
• Symmetry with respect to the origin means for every point \((a, b)\) on the graph, the point \((-a, -b)\) is also on the graph.

Domain and Range

Understanding the domain and range is crucial for working with functions. They define what values \(x\) can take and what \(f(x)\) can output. For the cube root function \(f(x) = \sqrt[3]{x}\), both the domain and range are all real numbers.

• Domain: Because you can take the cube root of any real number, whether positive or negative, the domain includes all real numbers. In interval notation: \((-\infty, \infty)\).
• Range: Similarly, the range of the cube root function is also all real numbers. This is because there is no restriction on the values that \(f(x)\) can produce. As \(x\) approaches positive or negative infinity, \(f(x)\) also becomes accordingly large or small. In interval notation: \((-\infty, \infty)\).

This means you can pick any \(x\) value from the real number set and find a corresponding \(f(x)\) value, and vice versa.

Real Numbers

Real numbers are a broad category of numbers including both rational and irrational numbers. Rational numbers can be expressed as a fraction of two integers, while irrational numbers cannot be written as a simple fraction.
Cube roots, like those we deal with in the function \(f(x) = \sqrt[3]{x}\), can produce any real number. For instance, the cube root of 8 is 2, which is a rational number. However, the cube root of a non-perfect cube (such as 9) is an irrational number, as it cannot be simplified into a neat fraction.
Real numbers are represented by {\(\mathbb{R}\)} and they are used to define both the domain and range for functions like the cube root function because they encompass all possible outputs and inputs. When dealing with cube roots, remember that any real number, whether rational or irrational, negative or positive, can be input to find \(f(x)\).