Precalculus

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Rules of Thumb

A function has no more than one output for any given input; if it has more than one, then it is not a function


What to Review

The graphs of polynomial functions
Graphs of other known functions


Definitions and Properties

Domain - the set of all values which can be operated on by a function to produce an output
Range - the possible output of a function
Composite functions: $f(g(x))$
Inverse functions:$ f(g(x))=g(f(x))=x$
Symmetric functions - functions can be symmetric with respect to the origin (odd functions) or with respect to the y axis (even functions). Inverse functions are symmetric over the line $y=x$
Periodic functions - functions such as $\sin\theta$ which repeat over a certain interval, or period
Monotonic functions - a function that is only decreasing or only increasing over an interval
Bounded functions - functions bounded by horizontal, vertical, or slant (oblique) asymptotes

Continuous functions - functions which satisfy the definition of continuity - a function is continuous at a point $c$ if and only if:

$f(c)$ exists
The $\displaystyle\lim_{x\to c}{f(x)}$ exists
This limit is equal to $f(c)$

Asymptotes

Vertical asymptotes: denominator of rational function cannot equal 0. If it does for any value of $x$, then the function becomes indeterminate at that point and there exists a vertical asymptote.
Horizontal asymptotes: 3 rules

If the degree of the numerator is less than the degree of the denominator, then the $x$-axis is the horizontal asymptote
If the degree of the numerator is equal to the degree of the denominator, then the ratio of leading coefficients is the horizontal asymptote
If the degree of the numerator is greater than the degree of the denominator, then there is no horizontal asymptote