Module 1 Review Essay examples

Module 1 Review
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

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