Precalculus Summary

Functions to know


Function properties

  Property Examples
Odd {$f(-x) = -f(x)$} {$x$}, {$x^3$}, {$x^{\frac{1}{3}}$}, {$\sin x$}, {$\tan x$}, {$\arcsin x$}, {$x\cos x$}, {$\frac{x^2+1}{x}$}
Even {$f(-x) = f(x)$} {$1$}, {$|x|$}, {$x^2$}, {$x^4$}, {$\cos x$}, {$x\sin x$}, {$e^{x^2}$}, {$\frac{\sin x}{x}$}
Neither   {$x+x^2$}, {$\ln x$}, {$e^x$}, {$\sqrt{x}$}


Domain problems

One-to-one and inverses

Polynomials and Rationals

Linear functions



Polynomial of degree {$n$}: {$f(x)=a_n x^n + a_{n-1} x^{n-1} + … + a_{1}x + a_{0}$}, {$a_n \ne 0$}

term As {$x\rightarrow \infty$} As {$x\rightarrow -\infty$}
{$f(x)=x^n+…\qquad$}, {$n$} even {$\infty$} {$\infty$}
{$f(x)=x^n+…\qquad$}, {$n$} odd {$-\infty$} {$\infty$}
{$f(x)=-x^n+…\qquad$}, {$n$} even {$-\infty$} {$-\infty$}
{$f(x)=-x^n+…\qquad$}, {$n$} odd {$\infty$} {$-\infty$}

Note how all the infinities invert when the coefficient was negative.

Zeros of polynomials

Long division and synthetic division

Rational functions

To analyze, {$f(x) = \frac{(x-1)(2x+3)(x-2)}{(x-3)^2(x-1)}$}

Exponentials and Logs

All the properties on the right column are gotten by taking the equations on the left column and taking the {$\log_a$} of both sides, reordering, and using {$X=a^x$} and {$Y=a^y$}.

Properties Exponential Log
Anything to the power of 0 is 1 {$a^0 = 1$} {$\log_a 1 = 0$}
Anything to the power of 1 is itself {$a^1 = a$} {$\log_a a = 1$}
Multiply powers ⇒ add exponents {$a^x a^y = a^{x+y}$} {$\log_a (X Y) = \log_a X+\log_a Y$}
Divide powers ⇒ subtract exponents {$\frac{a^x}{a^y} = a^{x-y}$} {$\log_a (\frac{X}{Y}) = \log_a X - \log_a Y$}
Inverses {$a^{\log_a X} = X$} {$\log_a a^x = x$}
Bring down the exponent {$a^{y \log_a X} = (a^{\log_a X})^y = X^y$} {$y \log_a X = \log_a (X^y)$}
Change of base {$a^x = (b^{\log_b a})^x$} {$\log_b X = \frac{\log_a X}{\log_a b}$}

You are not required to know the highlighted ones but they help you understand the bigger picture.


Trigonometric functions


function domain range
{$\arccos x = \cos^{-1} x$} {$ [-1, 1] $} {$ [0, \pi] $}
{$\arcsin x = \sin^{-1} x$} {$ [-1, 1] $} {$ [-\pi/2, \pi/2] $}
{$\arctan x = \tan^{-1} x$} {$(-\infty, \infty)$} {$(-\pi/2, \pi/2)$}

Trigonometric algebra

  Important ones Related ones
Pythagorean {$\cos^2 t + \sin^2 t = 1$} {$1 + \tan^2 t = \sec^2 t$}
{$\cot^2 t + 1 = \csc^2 t$}
Even/odd {$\cos(-t) = \cos t$}
{$\sin(-t) = -\sin t$}
{$\tan(-t) = -\tan t$}
similarly for {$\sec, \csc, \cot$}
Co-function {$\cos\left(\frac{\pi}{2}-t\right) = \sin t$}
{$\sin\left(\frac{\pi}{2}-t\right) = \cos t$}
similarly for {$\sec/\csc, \tan/\cot$}
Angle sum {$\cos(s+t) = \cos s \cos t - \sin s \sin t$}
{$\sin(s+t) = \sin s \cos t + \cos s \sin t$}
{$\cos(s-t) = \cos s \cos t + \sin s \sin t$}
{$\sin(s-t) = \sin s \cos t - \cos s \sin t$}
get {$\tan$} from {$\cos$} and {$\sin$}
Double angle {$\cos(2t) = \cos^2 t - \sin^2 t$}
{$\qquad= 2\cos^2 t - 1$}
{$\qquad= 1 - 2\sin^2 t$}
{$\sin(2t) = 2 \sin t \cos t$}
get {$\tan$} from {$\cos$} and {$\sin$}