Question

Consider the set \(f=\left\\{\left(x^{3}, x\right): x \in \mathbb{R}\right\\} .\) Is this a function from \(\mathbb{R}\) to \(\mathbb{R} ?\) Explain.

Short answer

Yes, the given set \(f=\{(x^3, x): x \in \mathbb{R}\}\) represents a function from \(\mathbb{R}\) to \(\mathbb{R}\), because for each x in \(\mathbb{R}\), there is exactly one x^3 in \(\mathbb{R}\).

Step-by-step solution

Identify the domain and codomain

From this relation \(f=\{(x^3, x): x \in \mathbb{R}\}\), we can see that the domain (which is x) is \(\mathbb{R}\). The codomain (which is f(x)) is also \(\mathbb{R}\). These are all possible x values and corresponding f(x) values.

Check if each element from domain has exactly one image in codomain

As a rule for functions, each element in the domain should map to exactly one image in the codomain. Here, for any given x in our domain, there exists exactly one x^3 in the codomain (\(\mathbb{R}\)). Hence, the given set \(f=\{(x^3, x): x \in \mathbb{R}\}\) fulfills this rule.

Final Conclusion

After analyzing both steps, we can conclude that for every x in the domain of the real numbers, there is exactly one image in the codomain. Therefore, the set \(f=\{(x^3, x): x \in \mathbb{R}\}\) does represent a function from \(\mathbb{R}\) to \(\mathbb{R}\).