Definitions, Notation, and Examples

Sets

A set is just a collection of objects. We don't care about ordering of the elements or repetition.

Examples of sets

a set of four letters.

a set of three numbers. Note that we don't care about the repetition, there are only three distinct elements of the set.

a set of infinitely many real numbers. The first definition of C is in set notation

Set notation uses braces to mark the beginning and end of the set. If the set just has a few elements they may be listed. Otherwise the set is divided into two halves, separated by a vertical bar. The first half names the set element and may also give you a larger set that the elements belong to. The second half gives conditions that the elements of the sets satisfy. The example above would be read "C is the set of all real numbers x such that x is greater than 2."

, the second is in interval notation.

Interval notation is used to express subsets of the real numbers as combinations (union, intersection, etc) of intervals. Intervals have two endpoints and include all the numbers between the endpoints. The endpoints themselves may or may not be part of the interval. Use a square bracket to indicate the an endpoint is in the interval. Use a parenthesis to indicate that the endpoint is not in the interval. Examples:

All the numbers between -1 and 1, not including -1, not including 1

All the numbers between -1 and 1, not including -1, including 1

All the numbers between -1 and 1, including -1, not including 1

All the numbers between -1 and 1, including -1, including 1
Additionally we can use infinity and -infinity with parentheses to indicate unbounded intervals, as in:

All the numbers greater than 1, not including 1.

Set operations and relationships

Symbol	Name	Description	Examples
	element of	An object is "an element of" a set, or "belongs to" the set. The symbol looks like an 'E' for 'Element.'	1 is an element of the set containing 1 and 2 3 is not an element of the set containing 1 and 2
	subset	One set is a subset of another, or is contained in another. The symbol is a rounded 'less than' symbol, indicating that all the elements of the "smaller" set are contained in the "bigger" set.	Every element of the first set also belongs to the second set, so the first set is a subset of the second. The first set contains the element 1, which is not in the second set, so the first set is not a subset of the second.
	union	Combines two sets into one new set by taking anything that belongs to either of the two sets. The symbol is like a 'U' for "union".
	intersection	Combines two sets into one new set by taking anything that belongs to both of the two sets. The symbol is like an 'n' for "iNtersection".
	set minus	Start with a set and remove the elements belonging to a subset.	Remove 1 and 2 from the set, leaving 3 and 4. Remove 0 from the real numbers, leaving all numbers except 0.

List of symbols for important sets

The "empty set". The set that contains no elements.

The "natural" or "counting" numbers.

The integers or the "whole" numbers.

The rational numbers. Those numbers which can be written as a fraction with integer numerator and denominator.

The real numbers.

Functions

A function is a relationship between two sets that assigns to each element of the input set a single element of the output set. The notation for this is

, which is read "f is a function from A to B." This indicates that A is the input set and B is the output set. In this course most of the functions we are interested in will be functions where the input set is some subset of the real numbers and the output set is the real numbers. Often these functions will be defined by a formula or by a graph. Given a function from a subset A of the real numbers into the real numbers, we can make a graph of the function by plotting all the points (a,f(a)) for a an element of A. This will give us some figure in the plane. On the other hand, given a figure in the plane, it is the graph of a function if and only if it passes the "vertical line test": any vertical line crosses the figure in at most one point.

How do I...?

Find the value of a function at a particular point?

If the function is given by a formula then evaluate the formula. If f(x)=2x+3, then to evaluate f at a point 'a' plug in an 'a' everywhere in the formula that you see an 'x'.
f(a)=2*a+3
f(2)=2*2+3=7
f(0)=2*0+3=3
f(4*a+b)=2*(4*a+b)+3=8a+2b+3
Notice in this last example that f "sees" the whole expression 4*a+b as its input, so all of (4*a+b) gets substituted in for the 'x'.

If the function f is defined by a graph then the value f(a) is the height of the graph above the point (a,0).

Find the domain of a function?

If the function is defined with notation such as

then the domain is the input set, A.

If the function is defined by a graph then the domain is all the x-coordinates which have a point of the graph directly above or below them.

If the function is defined by a formula the domain may be noted. For example consider the functions

and

The domain of f is specified to be [3,6]. Even though the formula defining f makes sense for values outside of [3,6], f is NOT defined for these values. You can not assume that you know what f ought to be outside of [3,6]. f and g are NOT equivalent functions because, for instance, g(0)=2, but f(0) is undefined, since 0 is not in the domain of f.

If the function is defined by a formula and no domain is specified then the domain is assumed to be all the real numbers for which the formula makes sense. Examples

Function	Domain	Explanation
		This function takes a real number and returns the number 2. This makes sense for any real number, so none need to be excluded.
		This function takes a real number and squares it. Again, this makes sense for all real numbers, so we don't need ot exclude any.
		The square root of a negative number is not defined, so we must restrict the domain to non-negative numbers.
		Division by zero is not defined, so we leave zero out of the domain, but we can divide by any other real number.
		We need to be careful whenever there is a quotient to exclude those x's which would make the denominator zero.
		We need to be carful whenever there is a square root that the quantity under the square root is non-negative. In this example that is true as long as x^2 is at least 1.

Find the range of a function?

The range is all the elements of the output set that actually get hit by the function. To show that a number b is in the range we must demonstrate an input a in the domain that will give us the output b.

If the function is defined by a graph then a number b is in the range if you can draw a horizontal line at height b and hit the graph.

If the function is defined by a formula then it may be possible to solve for the appropriate 'a'. Again, what we want to take a number 'b' and find some number 'a' such that f(a)=b. What we can try to do is let 'a' be an unknown and try to solve the equation f(a)=b for a. If we can find such an a in the domain then b is in the range, because we've found an element a of the domain that f sends to b.

Here is an example:
Find the range of

A number 'b' is in the range if there is an 'a' in the domain such that f(a)=b, so

multiply both sides by (a-1) to clear the denominator

distribute the b

move all the a's to the left

factor out the a

isolate a by dividing both sides by (1-b)

For b=1 this doesn't make sense because the denominator would be zero, but for any other 'b' this gives us a number 'a' such that f(a)=b, so the range is all reals except 1. More Examples

Function	Range	Explanation
	2	The function sends everything to the number 2, so 2 is the only thing that gets hit.
		0^2=0, so 0 is in the range. Any nonzero number squared will be positive, so we certainly don't have any negative numbers in the range. For any positive number b we could let a=sqrt(b). Then f(a)=f(sqrt(b))=(sqrt(b))^2=b.
		For any non-negative b let a=b^2. f(a)=sqrt(a)=sqrt(b^2)=\|b\|=b.
		For any non-zero b let a=1/b. Then f(a)=f(1/b)=1/(1/b)=b.
	[11,38]	f(3)=3^2+2=9+2=11. f(6)=6^2+2=36+2=38. For any b between 11 and 38 take a=sqrt(b-2). b-2 is between 9 and 36, so sqrt(b-2) is between 3 and 6, so a is in the domain. f(a)=a^2+2=(sqrt(b-2))^2+2=b-2+2=b.

Decide if a function is onto?

A function is "onto" if the range is the entire output set. See the previous question for how to determine the range.

Decide if a function is 1 to 1?

A function is 1 to 1 if every input gives a different output. Another way to say this is the numbers in the range only get hit once. A function is not 1 to 1 if you can find two different inputs that give the same output.

If you have a graph of the function, the function is 1 to 1 if it passes the horizontal line test: any horizontal line crosses the graph in at most one point. Here's an example of how you might show that a function is 1 to 1 algebraically:

We want to know if it is possible to evaluate f at some a and b and get the same answer.

cross multiply

distribute the a and the b

subtract ab from both sides

multiply both sides by -1

So the only way we can evaluate f at a and at b and get the same answer is if a=b. This shows that f is 1 to 1.

Decide if a function is even?

A function f is an "even function" if f(-x)=f(x).

If you have a formula for the function you can try evaluating f(-x) and see if you get the same answer as evaluating f(x). A function is not even if you can find any number a such that f(a) is different from f(-a).Here are some examples of even functions:
f(x)=2 is even because for any number 'a', f(a)=2=f(-a)
f(x)=|x| is even because for any 'a' f(-a)=|-a|=|a|=f(a)
f(x)=x is not even because f(-1)=-1 which is not equal to 1=f(1)
f(x)=x^2 is even because for any 'a' f(-a)=(-a)^2=(-1)^2*a^2=a^2=f(a)
f(x)=g(x)+h(x) is even if both g and h are even becuase then f(-a)=g(-a)+h(-a)=g(a)+h(a)=f(a)
f(x)=g(x)*h(x) is even if g and h are either both even or both odd
Note in the last example that multiplication of even functions behaves like addition of even and odd numbers. The product of two even or two odd functions is an even function, whereas the sum of two even numbers or two odd numbers is an even number.

Graphically even functions are symmetric about the y-axis. This means if you see a point of the graph on the right side of the graph you should see another point on the left side at the same height and the same distance from the y-axis. This is because if there is some point (a,b) on the graph then b=f(a), but if the function is even the f(-a)=f(a)=b, so (-a,b) should also be a point on the graph.

Decide if a function is odd?

A function f is an "odd function" if f(-x)=-f(x).

If you have a formula you can try evaluating f(-x) and see if you get the negative of f(x). A function is not odd if you can find a number a such that f(a) is different from -f(-a).Here are some examples of odd functions: f(x)=0 is odd becuase -f(-a)=-0=0=f(a)
f(x)=2 is not odd because -f(-1)=-2 which is not 2=f(a)
f(x)=x is odd because -f(-x)=-(-x)=x=f(x)
f(x)=x^2 is not odd because -f(-3)=-9 which is not equal to 9=f(3) f(x)=g(x)+h(x) is odd if both g and h are odd
f(x)=g(x)*h(x) is odd if one of g and h is even and the other is odd

Graphically we say that odd functions are symmetric about the origin. If there is a point (a,b) on the graph then b=f(a) so -f(-a)=f(a)=b, so f(-a)=-b, so the point (-a,-b) is also on the graph.

Hidden text

Back to Index of Notes