Elementary Functions
The most useful functions of real numbers (for us) will be functions which can be defined by some sort of a formula built out
of some small list of common functions, familiar from precalculus.
Examples of such functions include $f(x)=x^2-1$, $f(x)=\sin(x)$ and $f(x)=\log(x-3)$, along with more complicated
compositions such as
$$f(x)=\sqrt{\frac{\sin(x)+e^{\cos(3\log(\sqrt{2x+1}))}}{2-\tan(\cos({2\sin(x)+x^2-3x^3}))}}$$
We quickly review the building blocks of these functions, which will appear time and again throughout the course.
Power Functions
Power functions are defined by raising the variable (here, $x$) to some fixed real number power (here, $a$), written as $y=x^a$. The grap below allows you to investigate the behavior of this function as you vary the power $a$.
There are several special cases of power functions which will play an outsized role in calculus, and so we will go through these individually below.
Postive Integer Power These will be some of the most common functions you come across, including $x$, $x^2$ and $x^3$. They have natural domain the entire real line, and have their range equal to either the entire line (for odd exponents) or only the nonnegative numbers (for even exponents).
Negative Integer Power These are the reciprocal of positive integer powers; for any positive $a$, $x^{-a}=\frac{1}{x^a}$. Because $0^a=0$ for any nonzero $a$ and division by zero is undefined, these functions have natural domain containing all real numbers except zero.
Rational Power If $p/q$ is the ratio of two integers (with $q$ necessairly nonzero), we can describe the power function $x^{p/q}$ in terms of integer powers and roots. Recall that the $q^{th}$ root of a number $x$ is the number $y=\sqrt[q]{x}$ such that raising $y$ to the $q^{th}$ power returns $x$. Using this notion, we may express the function $x^{p/q}$ in the following equivalent ways: $$y=x^{p/q}=\sqrt[q]{x^p}=(\sqrt[q]{x})^p$$ The natural domain of such functions depends on both $p$ and $q$: if $q$ is even, then the domain is only the positive reals (as negative numbers to not have square roots, or $4^{th}$ roots, etc in the real number line). When $q$ is odd, the domain may either be the entire real line, except when $p$ is negative, where we must remove zero as previously.
Polynomials
A monomial is a single power function $x^n$ where the exponent is a positive integer: some examples are $x^2$, $x^6$, or $x^{32,403}$. The degree of a monomial is value of the integer exponent (the degree of $x^{14}$ is fourteen).
A polynomial is a sum of constant multiples of monomials; for example $f(x)=2x^5-3x^2+1$ or $g(x)=x^{14}-20x^3$. Each summand in a polynomial is called a term (so, $3x^2$ is a term in $x^3+3x^2-1$), the leading term is the term with the highest degree, and the degree of the polynomial is defined to be the degree of the leading term (so, $3x^3-x^{17}+x^2$ is a degree seventeen polynomial with leading term $-x^{17}$). The constant of each term is called the coefficient of that monomial (so, $3$ is the coefficient of $x^2$ in $3x^2+2$).
The natural domain of a polynomial is the entire real line: it makes sense to plug any real number into any monomial, and hence into any sum thereof. The range of a polynomial differs, depending on the polynomials degree: odd degree polynomials are onto functions, but even degree polynomials only output some subset of the real line (for example, $x^2$ only outputs nonnegative numbers.)
Rational Functions
Rational functions are just one step in complexity up from polynomials: instead of combining monomials only using addition, subtraction and multiplication, we now also allow division. The simplest definition is: a rational function is a quotient of two polynomials $r(x)=p(x)/q(x)$. For example, $$r(x)=\frac{3x^3-x^2+1}{5x^4-x^3+20x-1}$$
The natural domain of a rational function is the set of all real numbers except those that make the denominator zero (you can’t divide by zero!). Rational functions exhibit a myriad of different behaviors depending on the particular polynomials involved. Play around with the sliders in the Desmos below (or better, feel free to fully swap out the numerator and denominator with any polynomials of your choosing!) Can you make a rationa function that
- Has a vertical asymptote somewhere?
- Has a horizontal asymptote as it goes to infinity?
- Asymptotes to $y=x$ if you zoom out enough?
Exponentials
Exponential functions are an extension of the repeated multiplication to the entire real line. Instead of fixing the exponent and letting the base vary (as we did for power functions, above) exponentials fix the base, and treat the exponent as a variable. An exponential function $y=b^x$ is defined for every positive base $b$, and the natural domain of any of these exponentials is the entire real line. All exponentials intersect the $y$-axis at $1$ (as $b^0=1$ for all $b$), and have as their range all positive real numbers. Exponential functions are increasing when the base $b$ is greater than 1, and decreasing when $b<1$.
The fundamental properties of exponential functions, often called the laws of exponents are generalizations of the behavior of exponentiation-as-repeated-multiplication to arbitrary real number exponents:
$$b^xb^y=b^{x+y}$$
$$(b^x)^y=b^{xy}$$
While there is an exponential corresponding to every positive base, the most important by far is the exponential with base $e=2.718281828459…$. As we will see shortly, this exponential $y=e^x$ satisfies the very pleasing condition that its rate of change at a point is equal to its value at that point: which makes working with this base particularly efficient in calculus.
Logarithms
The logarithm is the inverse function to the exponential: that is, for a fixed base $b$, logarithms answer the question of for which $x$ does $b^x$ reach the value $y$? As an example: since $\log_2(8)$ is the answer to the question for which $x$ is $2^x$ equal to $8$? that is, $\log_2(8)=3$. Because exponential functions only output positive numbers, logarithms – their inverses – only take as input positive numbers, so the domain of logarithm functions is the postive number line. Logarithms have a vertical asymptote at $x=0$ heading towards negative infinity, and grow verrry slowly towards positive infinity, their range is the entire real line.
As the inverse of exponentials, logarithms satsify a collection of laws mirroring the laws of exponents. These are listed below:
$$\log_b(xy)=\log_b(x)+\log_b(y)$$ $$\log(x^y)=y\log_b(x)$$
The logarithm with base the natural number $e=2.7182818…$ is called the natural logarithm and gets its own notation $$\ln(x)=\log_e(x)$$
We can use the natural logarithm to express logs of all bases, using the relationship $\log_b(x)=\ln(x)/\ln(b)$ (note this relationship is not special to $\ln$, and any logarithm could be used in its place here: but we will find the natural choice by far the most convenient). Similarly, we can use the natural log to express any exponential functin $b^x$ in term of the natural exponential: $b^x=e^{x\ln b}$.
Trigonometric Functions
All trigonometric functions can be defined in terms of the ‘fundamental’ trigonometric functions $\cos(\theta)$ and $\sin(\theta)$, which measure the $x$ and $y$ components respectively of a line segment rotating around the unit circle (counterclockwise, starting on the $x$-axis).
How this process generates the graphs we recognize as the sine and cosine function is perhaps best seen in an animation:
Because we define these functions based on a rod that starts on the $x$-axis, the value of $\cos$ at $\theta=0$ is $1$ (the entire rod is on the $x$-axis!) and $\sin(0)=0$ (as a rod lying completely on the $x$-axis casts no shadow on the $y$ axis.) As we begin rotating the rod (as $\theta$ increases), we see that at first, $\cos\theta$ decreases, and $\sin\theta$ increases, as a larger portion of the rod becomes pointed along the $y$-axis. This behavior continues until $\theta=\pi/2$, when the rod is completely vertical, so $\sin(\pi/2)=1$ and $\cos(\pi/2)=0$, after which the rotating rod continues on its journey, and $\cos(\theta)$ becomes negative as $\sin(\theta)$ starts decreasing.
This description of the fundamental trigonometric functions in terms of a rotating line in a circle already tells us some fundamental information about their behavior:
Sine and Cosine are Periodic: Every $2\pi$ radians, or 360 degrees, a rotating line segment within a circle comes back to its original position. Thus, the values of $\sin(x)$ and $\cos(x)$ must return to their original values: $$\sin(x+2\pi)=\sin(x)$$
The Natural Domain of Sine and Cosine is the Entire Real Line: It makes sense to plug in any angle between $0$ and $2\pi$ into $\sin$ and $\cos$; as the result of the function just calculates the length of the shadow of a rotating rod placed at that angle. And, since the functions are $2\pi$ periodic, it then makes sense to plug in any real number; this just represents doing some whole number of turns before placing the rod at its eventual angle.
The Range of Sine and Cosine are Bounded: Since each of these trigonometric functions simply returns the $x$- or $y-$ coordinate of a point rotating around the unit circle, the outputs of both functions must always remain between $-1$ and $1$ (since the unit circle has radius $1$, by definition!)
Other Trigonometric Functions
In building more complex functions from these elementary pieces, we will often find ourselves dealing with expressions involving ratios of trigonometric functions, or fractions with a trigonometric function in the denominator. Due to the historical importance of trigonometry in navigation, many of these different situations were given specific names (for ease of reference), and some of these names have survived into the modern day. The most important such example is likely the tangent defined as $\tan(x)=\sin(x)/\cos(x)$: this function returns the height at which the line at $\theta$ radians intersects the (vertical)tangent to the circle at $(1,0)$.
These lines fail to intersect whenever the rotating rod itself is vertical. Thus, the natural domain of tangent is the set of real numbers which are not multiples of $\pi/2$ plus a multiple of $\pi$. As the rotating rod approaches vertical, its point of intersection with $x=1$ (that is, $\tan(\theta)$) diverges to $\pm\infty$, and the range of $\tan(\theta)$ is the entire number line.
Two other trigonometric functions which will show up time and again in calculus are the secant, and the cosecant functions. These are simply the reciprocals of $\cos$ and $\sin$ respectively: $$\sec(x)=\frac{1}{\cos x},\hspace{0.5cm}\csc(x)=\frac{1}{\sin x}$$
The natural domain of the cosecant is all real numbers except those where $\sin(\theta)=0$ (as this would make the denominator zero): so, everything except multiples of $\pi$. Similarly the natural domain of the secant includes all numbers except where $\cos(\theta)=0$ which first occurs at $\pi/2$, and then periodically every $\pi$ radians in each direction.
The range of secant/cosecant is also easy to determine: since $\sin$ and $\cos$ are never bigger in absolute value that $1$, their reciprocals are never smaller in absolute value than 1!
Inverse Trigonometric
If the trigonometric functions $\sin$ and $\cos$ return the $y$ and $x$ coordinates of a rotating line segment, their inverses must return the angle such a line segment must be rotated to in order to have a certain specified $x$ or $y$ coordinate. That is, $\arcsin(0.3)$ is the answer to the question “what angle must I rotate a rod in the plane so that its projection onto the $y$-axis is 0.5?"
Drawing the graph in Desmos above, we can see that the answer is given by an angle of $0.532$ radians, so $\arcsin(0.5)\simeq 0.532$. While clearly a useful notion, there’s an immediate practical difficulty: the trigonometric functions are not one-to-one, and so cannot have inverses! This issue manifests itself by the fact that the question “what angle must I rotate a rod in the plane so that its projection onto the $y$-axis is 0.5?" does not have a unique answer, but rather many: below is a second solution with a angle $\simeq 2.61$!
But its even worse than this: there are in fact infinitely many possible inverses of 1/2: take either of the above angles and add some whole number of full turns! To make a well-defined inverse function, we need to get rid of this ambiguity by choosing a domain which is small enough that we only get a single answer to questions such as the above.
For $\arcsin$, we see that for any solution given by a rod on the right-half of the circle, there is a ‘mirrored solution’ on the left half of the circle (like the pair shown above). Thus, if we restrict ourselves to the right half of the circle (to angles between $-\pi/2$ and $\pi/2$, we will have gotten rid of this ambiguity and be able to (finally!) define an inverse. The domain of arcsin is the set of possible $y$ values for points on teh unit cricle, so $[-1,1]$, and the range is the set of angles compriming the right half of the circle, $[-\pi/2,\pi/2]$
The inverse of cosine is defined similarly, by restricting the domain of $\cos$ to the top half (as opposed to the right half) of the circle. The domain of arccos is again the set of possible $x$-coordiantes on the unit circle, so the interval $[-1,1]$. But the range of arccos is now the angles $[0,\pi]$ defining the upper hemisphere. The graph of $\arccos(x)$ can be turned on in the above desmos widget for exploration.
The tangent function is also one-to-one on the left half of the unit circle (scroll up and play around with the widget defining tangent once more to convince yourself of this): thus, we can define an arctan function inverting tangent, with domain the entire real line (as $\tan(x)$ outputs every real number!), and range the angles $[-\pi/2,\pi/2]$. The graph of $\arctan(x)$ looks like a single connected component of the tangent graph flipped on its side, quickly approaching its horizontal asymptotes of $\pm\pi/2$.
We may also define inverses of the other derived trigonometric functions, such as $\sec$ and $\csc$, although these will be less useful to us overall. (Feel free to add them to any of the above desmos plots if you are curious, however).