Single Variable Calculus - Functions


A function can be visualized as a machine that takes in an input $x$ and returns an output $f(x)$. The collection of all possible inputs is called the domain, and the collection of all possible outputs is called the range.

Operations on Functions


the composition of two function, $f$ and $g$, is defined to be the function that takes as its input x and returns as its output $g(x)$ fed into f.


The inverse is the function that undoes $f$. if you plug f(x) into $f^{-1}$ you will get $x$. Notice that this function works both ways. If you plug $f^{-1}(x)$ into $f(x)$, you will get back $x$ again.

  • the $\arcsin$ function takes on values $[-\frac{\pi}{2},\frac{\pi}{2}]$ and has a restricted domain [-1,1].
  • The $\arccos$ function likewise has a restricted domain [-1,1], but it takes values $[0, \pi]$.
  • The $\arctan$ function has an unbounded domain, it is well defined for all inputs. But it has a restricted range $(-\frac{\pi}{2},\frac{\pi}{2})$

Classes of Functions


A polynomial $P(x)$ is a function of the form

The top power $n$ is called the degree of the polynomial. We can also write a polynomial using a summation notation.

Rational Functions

Rational functions are functions of the form $\frac{P(x)}{Q(x)}$ where each is a polynomial. You have to be careful of the denominator. When the denominator takes a value of zero, the function may not be well-defined.


Power functions are functions of the form $cx^{n}$, where $c$ and $n$ are constant real numbers.


Trigonometric Identities


The most common such function, referred to as the exponential, is $e^{x}$. This is the most common because of its nice integral and differential properties (below).

Algebraic properties of the exponential function:

Differential/integral properties:

Recall the graph of $e^{x}$, plotted here alongside its inverse, $lnx$

Note that the graphs are symmetric about the line $y = x$(as is true of the graphs of a function and its inverse).

Euler’s Formula

The Exponential

Properties of $e^{x}$

Polynomials are nice because they are easy to integrate and differentiate.

  1. $e^{x+y} = e^{x}+e^{y}$
  2. $e^{xy} = (e^{x})^{y} = (e^{y})^{x}$
  3. $\frac{\mathrm{d}}{\mathrm{d}x}e^{x} = e^{x}$
  4. $\int e^{k}\mathrm{d}x = e^{x} + C$

Consider the last two properties in terms of the long polynomial.Taking the derivative of the long polynomial for $e^{x}$ gives

More on the long polynomial

The idea of a long polynomial is reasonable, because it actually comes from taking a sequence of polynomials with higher and higher degree:

Each polynomial in the sequence is, in a sense, the best approximation possible of that degree. Put another way, taking the first several terms of the long polynomial gives a good polynomial approximation of the function. The more terms included, the better the approximation. This is how calculators compute the exponential function (without having to add up infinitely many things).

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