Question

Determine which of the following functions are onto: (a) \(F_{1}: \mathbb{R} \rightarrow \mathbb{R}\) where \(F_{1}(x)=x^{2}-1\). (b) \(F_{2}: \mathbb{R} \rightarrow \mathbb{Z}\) where \(F_{2}(x)=\lceil x\rceil(\lceil x]\) is the "ceiling" of \(x\) ). (c) \(F_{3}: \mathbb{Z} \rightarrow \mathbb{Z}\) where \(F_{3}(x)=x^{3}\). (d) \(F_{4}: \mathbb{R} \rightarrow \mathbb{R}\) where \(F_{4}(x)=x^{3}\). (e) For the linear ordering \(<\) on \(\mathbb{R}\), list all the increasing functions among parts (a) through (d). (f) For the ordering \(<\) on \(R\), list all the strictly increasing functions among parts (a) through (d).

Short answer

F2 and F4 are onto. F4 is strictly increasing. F4 is increasing.

Step-by-step solution

Determine if F1 is Onto

For a function to be onto, every element in the codomain must be mapped by some element from the domain. Consider function \(F_1(x) = x^2 - 1\). Let's see if for every \(y \in \mathbb{R}\), there exists an \(x \in \mathbb{R}\) such that \(x^2 - 1 = y\). Solving for \(x\), we have \(x^2 = y + 1\). For this to have real solutions, \(y+1\) must be non-negative, i.e., \(y \geq -1\). Since not every \(y \in \mathbb{R}\) satisfies this, \(F_1\) is not onto.

Determine if F2 is Onto

Function \( F_2(x) = \lceil x \rceil \) is from \(\mathbb{R} \to \mathbb{Z}\). For every integer \( z \in \mathbb{Z}\), \( F_2(z - 0.1) = z \) since \( \lceil z - 0.1 \rceil = z \). Thus, every integer is covered and \( F_2 \) is onto.

Determine if F3 is Onto

Consider \( F_3(x) = x^3 \) where \( x \in \mathbb{Z} \) and the codomain is \(\mathbb{Z}\). An integer \( y \) is in the form \( y = x^3 \). Checking values, for some \( y \), such as \( 2 \), there is no integer solution for \( x \) whereby \( x^3 = 2 \). Therefore, \( F_3 \) is not onto.

Determine if F4 is Onto

For \( F_4(x) = x^3 \) from \( \mathbb{R} \to \mathbb{R} \), every real \( y \) has the real solution \( x = \sqrt[3]{y} \) since the cube root of any real number is also a real number. Thus, \( F_4 \) is onto.

Increasing Functions in Real Numbers with Real Outputs

An increasing function retains the order \( x_1 < x_2 \Rightarrow f(x_1) \leq f(x_2) \). - \( F_1(x) = x^2 - 1 \) is not increasing due to its quadratic nature (it's not monotonic).- \( F_2(x) = \lceil x \rceil \) is piecewise constant, thus not increasing.- \( F_3(x) = x^3 \) (\(\mathbb{Z}\) to \(\mathbb{Z}\)) is not relevant but would be increasing.- \( F_4(x) = x^3 \) is increasing since the derivative \( 3x^2 \geq 0 \) all through \(\mathbb{R}\).

Strictly Increasing Functions

A strictly increasing function satisfies \( x_1 < x_2 \Rightarrow f(x_1) < f(x_2) \).- \( F_1(x) = x^2 - 1 \) is not strictly increasing.- \( F_2(x) = \lceil x \rceil \) is not strictly increasing.- \( F_3(x) = x^3 \) as \( F_4(x) = x^3 \) confirms \( F_3(x) = x^3 \) is not strictly increasing due to its integer nature.- \( F_4(x) = x^3 \) is strictly increasing as the function continuously strictly increases.

Key concepts

Real Analysis

Real analysis is a branch of mathematics that deals with real numbers and real-valued functions. It focuses on the properties and behavior of real functions, sequences, and series, along with their convergence and limits. Real analysis lays the foundation for understanding more complex mathematical theories such as calculus and differential equations.

In the context of functions, real analysis examines concepts such as continuity, differentiability, and integrals. To determine whether a function is onto, which means every element in the codomain is mapped by some element in the domain, real analysis tools are critical. For instance, evaluating if a polynomial such as a quadratic or cubic function maps every real number (domain) to another real number (codomain) is fundamental in real analysis. Functions like \(F_4(x) = x^3\) can be analyzed using their behavior and characteristics as set by the properties of real numbers to determine their properties, exemplifying the role of real analysis.

Increasing Functions

An increasing function is one where the order of the inputs is preserved in the output: if \(x_1 < x_2\), then \(f(x_1) \leq f(x_2)\). This subtle mathematical concept ensures that the function does not decrease as you move from left to right along the x-axis.

For instance, in the provided exercise, we assess whether each function follows this property within its defined domain and codomain. The function \(F_4(x) = x^3\), defined from \(\mathbb{R} \rightarrow \mathbb{R}\), is an increasing function because it progressively increases without decreasing, due to its derivative \(3x^2\) being non-negative across all real numbers. Understanding increasing functions is crucial for various applications in mathematics, including solving differential equations and optimization problems.

Strictly Increasing Functions

Strictly increasing functions take the idea of increasing functions a step further. In strictly increasing functions, not only is the order of inputs preserved, but also if \(x_1 < x_2\), then \(f(x_1) < f(x_2)\). This means the function never touches the same value again as you increase the input.

For instance, among the functions considered in the exercise, \(F_4(x) = x^3\) is strictly increasing when viewed over the real numbers because its derivative is always positive, ensuring a continual upward trend. This contrasts with piecewise constant or non-continuous functions, which can maintain the order but might step instead of continuously rising. In practical applications, strictly increasing functions are important in contexts where guarantees of progress, such as rate of return or growth models, are needed.