Section 2.2 Types of Functions
Subsection 2.2.1 Constant Functions
The simpliest kind of function is a constant function. Its output is the same regardless of input. The function \(f(x) =
5\) is constant at 5: it will give the value 5 no matter what value of \(x\) we specify. Constant functions have no domain restrictions.
Subsection 2.2.2 Linear Functions
Linear functions have the form \(f(x) = ax + b\) for real numbers \(a\) and \(b\text{.}\) Their graphs are straight lines, hence the name ‘linear’. Linear functions includes constant functions, since we allow \(a=0\text{.}\) All the tools from analytic geometry for understanding lines are useful for understanding linear functions. Linear functions have no domain restrictions.
A particularly important linear function is the function \(f(x)
= x\text{,}\) which is called the Identity Function. It is the unique function which takes any input and gives that input back without any action.
Subsection 2.2.3 Quadratic Functions
Quadratic functions have the form \(f(x) = ax^2 + bx + c\text{.}\) Their graphs are parabolas. We have a large array of tools to understand parabolas, including the vertex-form (to find the highest/lowest point of the function) and the quadratic equation (to find the roots). Quadratic functions have no domain restrictions.
Subsection 2.2.4 Polynomial Functions
I’ve used \(x\) in linear functions and \(x^2\) in quadratic functions. Allowing higher powers of the variable \(x\) leads to polynomial functions. Let me remind you briefly about polynomials. A polynomial is an expression that include whole positive powers of a variable, multiplied by numbers and added and subtracted together. This includes any expression like \(x^5 - 4x^3 + 2x^2 - 9x + 1\) or \(5x^2 +
19x - 25\text{.}\) The higest power in a polynomial is called the degree of the polynomial. A linear function is a degree one polynomial. A quadratic function is a degree two polynomial. Degree three polynomials are called cubics and degree four polynomials are called quartics. Any polynomial can be treated as a function, leading to the class of polynomial functions. If I want to write a very general form, I can do so as follows, where the \(a_i\) are any real numbers.
\begin{equation*}
f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_2x^2 + a_1 x +
a_0
\end{equation*}
Polynomial functions have a familiar standard shape involving a graph that curves up and down some number of times. The maximum number of times the graph of polynomial can change directions is one less than the degree. Figure 2.2.4 shows a cubic function; as a degree polynomial, it can change directions twice, which is precisely what the graph does. Polynomial functions have no domain restrictions.
Subsection 2.2.5 Rational Functions
Rational numbers are fractions involving integers, In the same way, rational functions are fractions involving polynomials. Rational functions have the follwoing form, where \(p(x)\) and \(q(x)\) are polynomials.
\begin{equation*}
f(x) = \frac{p(x)}{q(x)}
\end{equation*}
Rational functions may have domain restrictions. In order to avoid dividing by zero, I must avoid any \(x\) where \(q(x)
= 0\text{.}\) Rational functions may have vertical asymptotes near their undefined points. A vertical asymptote is a vertical line which the graph of the function approaches near an undefined point; they will be discussed in more detail in Subsection 4.1.2.
Subsection 2.2.6 Algebraic Functions
I am now moving into much broader categories of functions, where it is impossible to give a simple cohesive sense of the behaviour of the functions. However, terminology is still useful for grouping these functions. Algebraic functions are functions which involve the four basic operations of addition, subtraction, multiplication and division as well as any rational exponent. This includes polynomials and rational functions, but also roots, since roots can be written as fractional exponents. (\(\sqrt{x} = x^{\frac{1}{2}}\)). Algebraic functions can be very complicated conglomerations of these operations. They can have domain restrictions to avoid division by zero, as with algebraic function. They also have domain restrictions from the roots, since I can’t take an even root of a negative number. Any expression inside an even root must be positive.
Subsection 2.2.7 Trigonometric Functions
The first type of non-algebraic functions I want to present are trigonometric functions. Sine and cosine have the familiar sinusoidal wave shape: infinitenly many oscillations which repeat perfectly with some period. Sine and cosine waves can be analyzed by their amplitude and period. The other trigonometric functions (tangent, cotangent, secant and cosecant) are also periodic, but they have undefined points and vertical asymptotes at regular intervals.
Though the trigonometric functions are usually associated to triangles, it is more natural to define them using the circle. If \(\theta\) is the angle from the positive \(x\) axis, then \(\cos \theta\) is exactly the \(x\) coordinate of the point on circle determined by \(\theta\) and \(\sin
\theta\) is exactly the \(y\) coordinate. All of the important properties of trigonometric functions can be derived from circle geometry, including the many trigonometric identities. Please see the reference materials for definitions and identities involving trigonometric functions.
Subsection 2.2.8 Inverse Trigonometri Functions
Each trigonometric function has a inverse function. There are two standard notations; the inverse of \(\sin(x)\) is written either as \(\sin^{-1}(x)\) or \(\arcsin(x)\text{.}\) Even though the former notation is familiar from calculators, I will use the latter in these notes to avoid confusions with \(\frac{1}{\sin(x)}\text{,}\) which is a reciprocal, not an inverse. Each inverse trig function has its own specific domain; consult the reference materials for details. In Figure 2.2.9, I’ve drawn the graph of the inverse tangent function, \(f(x) = \arctan
x\text{.}\) It has a domain of all real numbers.
Subsection 2.2.9 Exponential and Logarithmic Functions
If \(a\) is a positive real number, then functions of the form \(f(x) = a^x\) are called exponential functions. They differ from roots and polynomials in that the variable is in the exponent. This distinction is very important but easy to confuse: \(f(x) = x^a\) is an algebraic function but \(f(x)
= a^x\) is an exponential function, which is not algebraic. The laws of exponents (see the reference material) are very useful for working with exponential functions.
The exponential bases you’ve most likely seen to date have been 2 and 10. These are useful bases, but in calculus (for reasons that will become clear later) a different base is prefered. The irrational number \(e\) is called Euler’s number. It has an approximate value \(e = 2.71828\ldots\text{.}\) It is by far the most common exponential base; you will be seeing the exponential function \(f(x) = e^x\) very frequently. It is reasonable to claim that \(f(x) = e^x\) is the most important function in calculus.
The inverse of the exponential function is the logarithm. If the exponential has base \(a\text{,}\) its logarithm is written \(\log_a x\text{.}\) However, the inverse of the exponential \(f(x) = e^x\) is instead written \(\ln x\) and called the natural logarithm. Like \(e^x\) over other exponential functions, the natural logarithm is by far the most commonly used logarithm in calculus. All logarithms have a domain restrictions: they can only act on (strictly) positive numbers.
Subsection 2.2.10 Hyperbolic Functions
Though I won’t define or investigate these function until Calculus II, it is useful to mention that there is a third important family of non-algebraic functions called the hyperbolic functions. There are a strange family, in some ways similar to trigonometric functions and in some ways similar to exponentials. They have inverse functions as well, called the inverse hyperbolics.
Subsection 2.2.11 Elementary Functions
Everything I’ve defined so far comes together to form the large family refered to as the elementary functions. Everything I’ve defined so far, as well as any combination of these functions, is an elementary function. These functions are sufficient for a wide variety of useful applications, and form the core of calculus and many other parts of applied mathematics. However, calculus also gives tools to create many other kinds of non-elementary functions. Depending how far you progress in the stream of calculus courses, you may encounter some of these non-elementary functions.
