Question

Let \(f: N \rightarrow N(N=\\{\) naturals \(\\})\). For each of the following functions, specify \(f[N]\), i.e., \(D_{f}^{\prime},\) and determine whether \(f\) is one to one and onto \(N,\) given that for all \(x \in N\) (i) \(f(x)=x^{3} ;\) (ii) \(f(x)=1 ;\) (iii) \(f(x)=|x|+3 ;\) (iv) \(f(x)=x^{2}\) (v) \(f(x)=4 x+5\). Do all this also if \(N\) denotes (a) the set of all integers; (b) the set of all reals.

Short answer

(i) f(x)=x^3: Injective, surjective for reals; (ii) f(x)=1: Not injective or surjective; (iii) f(x)=|x|+3: Not injective, surjective for [3,∞); (iv) f(x)=x^2: Not injective, surjective for naturals; (v) f(x)=4x+5: Injective, surjective for integers and reals.

Step-by-step solution

Understanding the Function Types

Before we determine the set of outputs or codomain and whether the functions are one-to-one and onto, it's important to understand the definitions: - **One-to-One (Injective):** Each element of the domain maps to a distinct element of the codomain. - **Onto (Surjective):** Every element in the codomain has a preimage in the domain. We'll apply these definitions to each function across different specified domains: naturals, integers, and reals.

Analyze f(x) = x^3

- **Domain:** When \(N\) is natural numbers, \(D_f' = \{1, 8, 27, \ldots\}\), but when \(N\) includes all integers, all cubes, positive and negative will be outputs, and for reals, all real numbers are outputs.- **One-to-One:** The function is injective because different inputs produce different cubes.- **Onto:** Only onto in the real case, as any real number has a cube root.

Analyze f(x) = 1

- **Domain:** Regardless of the domain (naturals, integers, reals), \(D_f' = \{1\}\).- **One-to-One:** This function is not injective as every input maps to the same value.- **Onto:** Not surjective unless the codomain is restricted to \{1\}.

Analyze f(x) = |x| + 3

- **Domain:** For naturals, \(D_f' = \{3, 4, 5, \ldots\}\); for integers, it's the same, and for reals, \([3, \infty)\).- **One-to-One:** Only for non-negative integers and reals because different values can map to the same output; otherwise, not injective.- **Onto:** Onto if codomain is chosen starting from 3.

Analyze f(x) = x^2

- **Domain:** When \(N\) is naturals, \(D_f' = \{1, 4, 9, \ldots\}\); all square numbers. For integers, \(D_f' = \{0, 1, 4, 9, \dots\}\), and for reals \([0, \infty)\).- **One-to-One:** Not injective over integers or reals because two distinct numbers can yield the same square (e.g., -2 and 2).- **Onto:** This is surjective for naturals starting from 0 to infinity.

Analyze f(x) = 4x + 5

- **Domain:** For natural numbers, \(D_f' = \{9, 13, 17, \ldots \}\); for integers, all integers are possible since multiplying by 4 and adding 5 covers whole integers; for reals, all real numbers are outputs.- **One-to-One:** This function is injective because each input yields a distinct output.- **Onto:** Functional only onto for reals and any kind of domain identical with these transformations (i.e., multiples of 4+5 evolving on a similar set).

Key concepts

One-to-one (injective)

A function is considered one-to-one, or injective, if every element in the domain maps to a unique element in the codomain. This means that no two different inputs will produce the same output.

• Distinct Outputs: For a function to be injective, each distinct input must map to distinct outputs. So, if \(f(a) = f(b)\), then \(a\) must equal \(b\).
• Example: Consider the function \(f(x) = x^3\). For real numbers, this function is injective because different values of \(x\) (including negative and positive values) will produce different cubes, making it impossible for two different values of \(x\) to result in the same \(x^3\).

Recognizing injective functions helps understand how data transforms and ensures no overlaps in mapping from inputs to outputs.

Onto (surjective)

A function is onto, or surjective, if every element of the codomain is an image of at least one element from the domain. This implies that the function covers the entire codomain.

• Comprehensive Mapping: For a function to be surjective, for every \(y\) in the codomain, there must be at least one \(x\) in the domain such that \(f(x) = y\).
• Example with \(f(x) = x^3\): Over the reals, this function is surjective because every real number has a cube root. Therefore, for any real \(y\), you can find an \(x\) (which is \(\sqrt[3]{y}\)) such that \(f(x) = y\).

Understanding surjective functions is essential for mapping all possible outputs, ensuring that the function fully utilizes its codomain.

Real numbers

The set of real numbers is an incredibly important concept in mathematics. It includes all the numbers on the continuous number line, both rational and irrational.

• Definition: Real numbers encompass a broad range including natural numbers, integers, fractions, and numbers like \(\pi\) and \(\sqrt{2}\) that can't be expressed as simple fractions.
• Continuous Nature: Unlike discrete sets, the real numbers form an unbroken line. Every point on this line corresponds to a real number, allowing for seamless mapping in functions covering the reals.
• Relevance to Functions: When analyzing function behavior, especially in continuity and smoothness, the domain being real numbers can significantly change the properties, such as being onto. For example, in the case of \(f(x)=x^3\), including negative, zero, and positive values ensures it maps to all real numbers.

Working with real numbers means engaging with diverse mathematical problems and solutions, making them foundational in math, science, and engineering.