Problem 1 - A Gentle Introduction to the Art of Mathematics

Question

For each of the following functions, give its domain, range and a possible codomain. (a) \(f(x)=\sin (x)\) (b) \(g(x)=e^{x}\) (c) \(h(x)=x^{2}\) (d) \(m(x)=\frac{x^{2}+1}{x^{2}-1}\) (e) \(n(x)=\lfloor x\rfloor\) (f) \(p(x)=\langle\cos (x), \sin (x)\rangle\)

Step-by-Step Solution

Answer

Domains: (a) \mathbb{R}\, (b) \mathbb{R}\, (c) \mathbb{R}\, (d) \mathbb{R} - \{\pm1\}\, (e) \mathbb{R}\, (f) \mathbb{R}\. Ranges: (a) \[-1, 1\], (b) \(0, \infty\), (c) \[0, \infty\], (d) \((-\infty, -1)\cup(1, \infty)\), (e) \mathbb{Z}\, (f) \{(a,b)|a^2+b^2=1\}. Possible Codomains: (a) \mathbb{R}\ or \[-1, 1\], (b-c) \mathbb{R}\, (d) \mathbb{R}\, (e) \mathbb{R}\, (f) \mathbb{R}^2\.

Step 1: Determine the Domain for Each Function

For each function, determine the set of input values (x-values) for which the function is defined. (a) For the sine function, \(f(x)=\sin(x)\), the domain is all real numbers, \(-\infty < x < \infty\). (b) For the exponential function, \(g(x)=e^{x}\), the domain is again all real numbers, \(-\infty < x < \infty\). (c) For the quadratic function, \(h(x)=x^{2}\), the domain is all real numbers, \(-\infty < x < \infty\). (d) For the rational function, \(m(x)=\frac{x^2+1}{x^2-1}\), the function is not defined at points where the denominator is zero, i.e., when \(x^2=1\) so the domain is all real numbers except \(x=\pm1\). (e) For the floor function, \(n(x)=\lfloor x \rfloor\), the domain is all real numbers, \(-\infty < x < \infty\). (f) For the function, \(p(x)=\langle\cos(x),\sin(x)\rangle\), the domain is all real numbers, \(-\infty < x < \infty\).

Step 2: Determine the Range for Each Function

For each function, determine the set of output values (y-values) that the function can take. (a) The range of the sine function, \(f(x)=\sin(x)\), is \([-1, 1]\). (b) The range of the exponential function, \(g(x)=e^{x}\), is \((0, \infty)\). (c) The range of the quadratic function, \(h(x)=x^{2}\), is \([0, \infty)\). (d) The range of the rational function, \(m(x)=\frac{x^2+1}{x^2-1}\), is \((-\infty, -1)\cup(1, \infty)\). (e) The range of the floor function, \(n(x)=\lfloor x \rfloor\), is the set of all integers, \mathbb{Z}\. (f) The range of the function, \(p(x)=\langle\cos(x),\sin(x)\rangle\), is the unit circle, \{(a, b)|a^2 + b^2 = 1\}\.

Step 3: Determine the Codomain for Each Function

The codomain of a function is the set into which all the outputs of the function are constrained to fall. This is similar to the range but it is defined by the problem's context. (a) For the sine function, \(f(x)=\sin(x)\), a possible codomain is \([-1, 1]\) though it can also be \mathbb{R}\. (b) For the exponential function, \(g(x)=e^{x}\), a possible codomain is \mathbb{R}\. (c) For the quadratic function, \(h(x)=x^{2}\), a possible codomain is \mathbb{R}\. (d) For the rational function, \(m(x)=\frac{x^2+1}{x^2-1}\), a possible codomain is \mathbb{R}\. (e) For the floor function, \(n(x)=\lfloor x \rfloor\), a possible codomain is \mathbb{R}\. (f) For the function, \(p(x)=\langle\cos(x),\sin(x)\rangle\), a possible codomain is \mathbb{R}^2\.

Key Concepts

Function Analysis

In mathematics, function analysis is essential for understanding how different types of functions work. Let's break down the key components one by one:

• Domain: The set of all possible input values for which the function is defined. For example, the domain for the sine function, \(f(x) = \text{sin}(x)\), is all real numbers (\(-\infty \text{ to } \infty\)).
• Range: The set of all possible outputs a function can produce. For instance, the range of the sine function is \([-1, 1]\).
• Codomain: The set that includes all possible outputs of a function. This is usually provided by the context or problem. For the sine function, the codomain can be all real numbers, though it is typically restricted to \([-1, 1]\).

Understanding these three components helps in the correct analysis and representation of any function. Each one provides a different perspective on how functions map inputs to outputs. By analyzing the given functions, you can then understand their behavior and limitations.

Codomain

The codomain of a function is an important concept that often gets confused with the range. Unlike the range, which is the set of all actual outputs a function can produce, the codomain is the set that contains all potential outputs. This is determined by the context or the definition given in a problem.

• For \(f(x) = \text{sin}(x)\), the codomain is usually \([-1, 1]\).
• For \(g(x) = e^{x}\), the codomain can be all real numbers, although its range is \((0,\infty)\).
• For the quadratic function \(h(x) = x^2\), the codomain can also be all real numbers.

The distinction between codomain and range might seem trivial. However, it's particularly useful in more advanced topics like injective and surjective functions. Always keep in mind that the codomain is a broader set into which the range fits.

Mathematical Functions

Mathematical functions are as varied as the many different branches of mathematics. Understanding their types and characteristics is foundational for advanced studies. Here, we'll outline a few types and their characteristics.

• Sine Function \(f(x) = \text{sin}(x)\): Periodic and oscillatory with domain \((-\infty, \infty)\) and range \([-1, 1]\).
• Exponential Function \(g(x) = e^{x}\): Continuous and monotonic with domain \((-\infty, \infty)\) and range \((0, \infty)\).
• Quadratic Function \(h(x) = x^2\): Parabolic with domain \((-\infty, \infty)\) and range \([0, \infty)\).
• Rational Function \(m(x) = \frac{x^2+1}{x^2-1}\): Excludes points where the denominator is zero (\(x= \pm1\)). Its range is \((-\infty, -1) \cup (1, \infty)\).
• Floor Function \(n(x) = \lfloor x \rfloor\): Steps down to the nearest integer with domain \((-\infty, \infty)\) and range of all integers (\mathbb{Z}).
• Vector-valued Function \(p(x)=\langle\text{cos}(x), \text{sin}(x)\rangle\): Represents points on the unit circle with domain \((-\infty, \infty)\) and range where \(a^2 + b^2 = 1\).

The variety within mathematical functions is immense, but understanding their types and properties is crucial to mastering the subject. Each type of function brings unique characteristics and behaviors that are essential for problem-solving and mathematical proofs.