Vertical Line Test

If a vertical line can be drawn through more than two points on the graph of a relation, then the relation is not a function.

Horizontal Line Test

If a horizontal line can be drawn through more than two points on the graph of a relation, then the relation is not one-to-one.

Domain

The domain of a function is restricted in the following cases.

1. The denominator may not equal zero.

2. Any value under a square root must be at least zero.

Range

To prove that this is in fact the inverse of our original f(x), use composition f[f-1(x)] and f-1[f(x)] to produce x in each case.

Even Function

If x is replaced with a –x and the function remains the same, then the function is even and is symmetric with respect to the y axis.

Odd Function

If x is replaced with –x and all terms in the function return the opposite sign, then the function is odd and is symmetric with respect to the origin.

Definition of Increasing x1 < x < x2 implies f(x1) < f(x2) A function f is increasing on an interval if, for any x1 and x2 in the interval,

x1< x2 implies f(x1) < f(x2).

That just means that as the x values increase, so do the corresponding y values.

Definition of Decreasing

A function f is decreasing on an interval if, for any x1 and x2 in the interval,

x1< x2 implies f(x1) > f(x2).

This means that as x goes up, y goes down!

Definition of Constant Functions

A function f is constant on an interval if, for any x1 and x2 in the interval,

f(x1) = f(x2)

This means that the y coordinates stay the same.

Relative Minimum

A function value f(a) is called a relative minimum of f if there exists:

an interval (x1, x2) that contains a such that

x1 < x < x2 implies f(a) f(x).

We see this as a "low" point in this interval.

Relative Maximum

A function value f(a) is called a relative maximum of f if there exists an interval (x1, x2) that contains a such that

x1 < x < x2 implies f(a) f(x).

Relative maximums look like local "high" points in the graph.

Compositions of Functions

The composition of the function f with g is

(f ° g)(x) = f(g(x))

The domain of (f g) is the set…