Question

Consider the set \(f=\left\\{\left(x^{2}, x\right): x \in \mathbb{R}\right\\}\). Is this a function from \(\mathbb{R}\) to \(\mathbb{R}\) ? Explain.

Short answer

Yes, set \(f\) is a function from \(\mathbb{R}\) to \(\mathbb{R}\).

Step-by-step solution

Identify the rule

The rule, as described in set \(f\), assigns to each real number \(x\) its square \(x^{2}\). This rule is commonly known as 'squaring'.

Verify the criterion for being a function

In a function, every element in the domain (in this case, \(\mathbb{R}\)) must be associated with exactly one element in the codomain (also \(\mathbb{R}\)). The square of any real number \(x\) is also a real number since \(\mathbb{R}^{2}=\mathbb{R}\). Thus, every \(x \in \mathbb{R}\) is associated with exactly one \(x^{2} \in \mathbb{R}\).

Conclude if set \(f\) meets the criteria

Since for every input \(x \in \mathbb{R}\), there exists a unique output \(x^{2} \in \mathbb{R}\), set \(f\) indeed meets the criteria to be a function from \(\mathbb{R}\) to \(\mathbb{R}\).

Key concepts

Domain and Range

Understanding the domain and range of a function is crucial in mathematics. The domain of a function consists of all the possible values that can be input into the function, which are called 'inputs' or 'arguments.' Conversely, the range is made up of all possible outputs or results that the function can produce after applying it to the domain.

In the context of the exercise, the domain of the function with the rule that assigns square values to real numbers is the set of all real numbers, denoted as \( \mathbb{R} \). The range is also within the real numbers since squaring any real number results in another real number. However, it's worth noting that the range in this particular case does not include negative numbers, as squares of real numbers are always non-negative.

Real Numbers

The set of real numbers, symbolized by \( \mathbb{R} \), includes all the numbers that can be found on the number line. This includes all the rational numbers, like fractions and integers, as well as the irrational numbers, such as \( \sqrt{2} \) or \( \pi \).

They are essential in the study of continuous phenomena and appear frequently in various fields of mathematics and science. In the function from the exercise, \( \mathbb{R} \) represents both the domain and codomain, signifying that you can put in any real number and expect to get a real number out after performing the squaring operation.

Mathematical Functions Criteria

For a set to be considered a function in mathematics, it must fulfill certain criteria. Each input from the domain should be associated with exactly one output in the codomain. If an input corresponds to more than one output, then the set does not define a function.

Following the steps outlined in the textbook's solution, set \( f \) indeed qualifies as a function because every real number \( x \), when squared, yields a unique result \( x^2 \), ensuring that there is a single, well-defined output for each input. This is known as the 'Vertical Line Test' in graphical representations, where a vertical line intersects the graph of the function at most once. Consistency according to these criteria ensures we can reliably work with functions in mathematical operations and problem-solving.

Squaring Operation

In mathematics, the squaring operation takes a number and multiplies it by itself. This is represented as \( x^2 \), which is read as 'x squared.' When you square any real number, whether it is positive, negative, or zero, the result is always non-negative.

This is due to the fact that a negative number multiplied by itself results in a positive number. For instance, \( (-3)^2 = (-3) \times (-3) = 9 \). Squaring is a fundamental operation that forms the basis for more complex mathematical concepts like power functions and quadratic equations. It is key to understanding the behavior of parabolic graphs, which visually represent the solutions to quadratic functions.