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## Rational Functions and Other Essential Functions

Part two of the Essential Functions series examines power, trigonometric and rational functions. Power functions have the form of x^a, where a can be any real number. This part of the video series examines cases 2 (a between 0 and 1) and case 3 (a less than 0). The first case is examined in part 1.

The power functions are graphed for each case to visualise the power relationship. Case 3 functions will create infinite discontinuities (asymptotes). Rational functions are formed by the division of two polynomial functions; P(x) and Q(x). Infinite discontinuities will occur wherever Q(x)=0.

This is because any number divided by 0 is essentially infinite. Vertical asymptotes will form wherever the bottom of the function is nothing. The number of discontinuities depends on the order of Q(x). Trigonometric functions have been studied in algebra but will be needed for future calculus studies.

Sine, cosine, and tangent functions are periodic. Sin(x) and Cos(x) have a period of 2*pi whereas tan(x) has a period of pi. Sin(x) looks like a shifted version of Cos(x), they are essentially the same but are 90 degrees out of phase. The range of sine and cosine is between -1 and 1. The range of the tangent function is the whole real number set. The trigonometric functions are graphed so that they can be visualized and memorized. Until next lessons guys!